Virtual Mindfulness Retreat with Khan Academy and Headspace

WEBVTT
Kind: captions
Language: en

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- The intention for today's
hour is really just to relax

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just to unwind, not a lot of
information coming at you,

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just embodied practices.

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And I know that a lot of you
probably have commitments

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at home right now, maybe kids coming in.

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And so really just do what you can

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and take those breaks as needed

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and really there's no pressure to do this

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in any certain way.

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So we're just gonna start with
something that helps us relax

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and that's stretching and
a little bit of massage.

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So Jeremy, if you wouldn't
mind, we're gonna just have

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some background music while we do this.

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And wherever you are, if
you're seated at a chair

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or I'm sitting on the floor right now

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'cause that helps me relax a little more

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but wherever you are,
if you're comfortable

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you can just close your
eyes just to shut off

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some stimuli that's unneeded right now.

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It might be nighttime or
evening where you are,

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so it might be cooler to
just take a moment to notice

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the temperature and the
environment around you.

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And then just take a moment
to notice the feeling

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of your body on the chair,
the surface that you're on,

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and notice where your hands are resting,

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maybe notice if you're leaning
forward or leaning back,

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or if there's any adjustment
that you'd like to make.

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So there's more ease in your body.

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And we're gonna set the
intention of letting go of time

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and that sounds like a giant task,

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but we'll see if we can touch

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into this sense of timelessness

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as if we had all the time in the world

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to let go and unwind.

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Just take a moment to
connect with your breath

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breathing in through your nose

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(inhales)

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and then breathing out a little slower

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through your nose or mouth,

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just letting our nervous
systems start to unwind

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and then just let your
breath find its own natural

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easy rhythm, letting go of any effort

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and then just notice any
sensations in your body.

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So you might be hungry, you might be full,

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you might notice your eyes are fluttering

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and just as we notice these sensations

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we don't have to try and change
them or make them different

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but we can just kind of
greet them with curiosity

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and acceptance, and that this
is how things are right now

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and that's okay.

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If you happen to notice pain

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it might be difficult to
stay with that sensation

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so you can just touch
into any sensation of pain

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and then find a place in
your body where there's ease.

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And sometimes we really
have to search for ease

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maybe it's at the tip of our big toe

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or one of our thumbs,
but somewhere in the body

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usually there's a contact with ease.

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So just allowing us to be
as we are in this moment,

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less than perfect, just as we are.

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And we're just going to add in
a little bit of movement here

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just to stretch our body gently

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and if any of these
movements don't work for you

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you can just stay resting

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and whatever position is comfortable.

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So if you'd like to open your
eyes to see these stretches

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you can, or you can keep your eyes closed

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and just follow my verbal instructions.

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So go ahead and anchor your
hand to a spot next to you

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maybe you're holding onto a chair

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or your fingertips are
resting on a couch or a desk

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and then we're gonna lift our left arm up,

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lift our left fingertips toward the sky

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and just feel that length all the way down

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from your fingertips to your hip.

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And then we're just gonna
slowly reach over to our right

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you can look down that
is nice on your neck

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or you can look up and I'm
just keeping my eyes closed

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just to stay embodied, just to stay with

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the pleasant sensations of stretch

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and loosening up the muscles.

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Take a deep breath here
feeling your lungs expand

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and a deep breath out,

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and then reaching back up to the center

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and then just letting
that left arm float down

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and just pause to notice
different sensations

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between the side we stretched

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and the one we haven't stretched.

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*Subtle tingling or heat

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maybe more space in your
diaphragm to breathe.

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And then we'll go to the other side

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so just lifting that right
arm up toward the sky

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reaching the fingertips up,
anchoring your left hand

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then reach up and over toward your left,

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feeling that stretch from
your right hip all the way

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through to your right fingertips

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and again, you can look
down, you can look up

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and just take a few deep breaths
feeling your lungs expand.

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And then on your next in
breath just reaching back up

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and as you exhale slowly letting
that right arm drift down

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again, just pausing here

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noticing if you feel a little
different than when we started

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maybe you can feel the nervous
systems start to unwind.

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And if not, that's okay too,

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sometimes our nervous system
kind of holds on tight

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to what's going on or
what kind of day we had.

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So that's okay too.

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And then you can open your
eyes and just follow along

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we're just going to do a few movements

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to give ourselves some love,
give our muscles some love.

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So just go ahead, if
it's comfortable for you

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and you don't have any injuries
just go ahead and squeeze,

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gentle squeeze here, right
in your shoulder blades.

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This is where we hold a lot of tension

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and then down the arm all
the way down to your forearm

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and give a little squeezed
her hand and your fingers,

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and let's just do that one
more time going back up,

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squeezing and then again, pause,

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just to notice the difference
between the side we squeezed

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and the side we haven't yet.

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A head, it's all about awareness, right?

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We're just bringing awareness
to ease so we can identify it.

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And then let's go to the other side

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so just giving it a nice
squeeze to that shoulder blade

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anywhere you feel tight
you can stay in that area

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if you'd like and then you
can just work your way down

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the arm, giving a squeeze to
the shoulder and the forearm

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all the way down to your hand.

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Maybe just look down at
your hand and your fingers

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thinking about all the
things that they do,

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they type, and they hug,
cook dinner, wash dishes

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just giving love.

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And then again, just pausing and seeing

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if you can really feel
into the sensations,

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maybe tingling, maybe lightness,
maybe some ease emerged

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then we're gonna do one last movement.

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So we're gonna make fists
with our hands like this,

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you can tuck the thumbs
and do your fingers

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and just hold the squeeze
for a few moments.

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Noticing what happens in
the mind is we tense up

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this is kind of representative
of how we sometimes

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hold on to things or
resist what's happening

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and then take a deep breath
in through your nose.

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(inhales)

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And as you breathe out,
just release the hands

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opening the palms and
this hand gesture kind of

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represents letting go or accepting,

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or being open and curious,
notice how different that is.

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And then the last movement we'll do

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if you wanna open your
eyes to look is just

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imagine you're kind of hugging a tree

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and relax the shoulders down.

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We don't want them to
be up toward your ears.

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And imagine this gesture,
this posture is like

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we're holding whatever we
encounter in this next 40 minutes

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together with kindness,

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whether someone interrupts our
session or we don't feel it,

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we don't feel what we think
we're supposed to be feeling.

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See if we can hold all of that
with kindness and compassion.

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And then we can rest our hands
back down, opening our eyes.

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So hopefully the nervous
system has started to enter

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what we call the rest and
digest mode of operating, right?

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Our parasympathetic nervous system.

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The next few activities we're
gonna do are all about that.

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All about activating our
parasympathetic nervous system

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which is the opposite of our
sympathetic nervous system

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which is our, what we call
fight or flight, right?

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When we're stressed or we're
trying to get things done

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or we're rushed, we wanna
move into the other way,

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the opposite of that.

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So I have a little exercise,

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which is about thinking about teachers

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that you've had in your life.

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And these don't have to be
teachers you've had at school

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but any kind of teacher that's
left an impression on you,

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someone that has taught you
something or connected with you

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someone with whom you felt really seen.

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And I want you just to reflect on that

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and maybe a few people
could share what comes up

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when you think about that.

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And as you're thinking,
as you're processing,

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I'll just share a person I thought of.

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She was my 11th grade science teacher

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and I was really disenchanted with school,

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at the time, I felt like I
wasn't able to really focus on

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what I wanted to focus on,

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which was neuroscience and psychology.

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And she happened to go to UCLA

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which at the time was my dream school

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and she gave me a textbook from one

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of her neuroscience
classes that she still had

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and she told me that at
any time I could go sit

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in *on classes at UCLA.

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And I was so appreciative of
that because first of all,

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she took my interests seriously,

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she gave me a book that was special to her

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and she let me know that I
could pursue my interests

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and I didn't have to wait for it,

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I could go sit in on a
lecture and be part of it

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even though I hadn't
graduated high school yet.

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So that was a teacher
that I'll never forget,

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I'll remember her for the rest of my life.

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So is there anyone who'd like to share

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just a story of a teacher
you can share verbally,

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I would love to hear your voices,

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and I totally understand too,

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if you you're not able to share verbally

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'cause of what's going on at home.

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But yeah, who's a teacher
in your life that you've had

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maybe it's even a teacher
who inspired you to go

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into this profession and yeah-

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- [Jeremy] We're getting a
number of great responses

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in the question section, which is awesome.

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Carrie says her ninth to
12th grade music teacher

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was totally present and authentic,

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such an encouragement to all students.

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And Katie says, fifth
grade teacher that I worked

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with when I was a para,

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I still think about him
and my classroom plans.

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And then just to get a couple
of folks to share live,

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I see some folks have raised their hands

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so I'm gonna actually
unmute crystal Davis here.

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So crystal, if you like,
you are now live with Sam.

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- Hi crystal.

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- Sometimes it can take a little while

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for the audio to kick it.

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Crystal, feel free to share,

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I'm also gonna go over to Theresa here.

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Theresa, if you're there,
feel free to share.

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- [Crystal] Okay, they said I was muted,

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can you hear me now?

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- Yeah, we can hear you-
- Pretty great, yeah.

00:15:25.650 --> 00:15:27.150
- [Crystal] Okay, awesome.

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So for me, it's my fifth grade teacher,

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her name was Gail Lidy,

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I remember her like it was yesterday.

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I had a very challenging childhood,

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I'll just leave it at that

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to make it comfortable for everyone

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but she showed me that she loved me

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even though I felt so uncomfortable.

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And the biggest thing of all is that

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I was actually able to meet her

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after I finally got my
degree six years ago

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and I did my student teaching

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and she told me how
proud of me that she was.

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And so I got a chance to
have that last conversation

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with her before she passed away,

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so just very super precious.

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That's my share.

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- That's amazing.

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Thank you for sharing that

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and I heard that the word
love really resonated with me.

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Like you know when you're loved,

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you know when you're cherished by someone,

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even if it is a teacher,
that kind of connection

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can translate or transcend to love

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and that's why we're here,

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I would say that's why
we're here on earth.

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You know, we're here to
connect, we're here to love.

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So it's an amazing connection you had

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and so great that you
were able to meet with her

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and let her know what
impact she had on you

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because when we're kids

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and my husband just told
me this too, he's like

00:16:46.040 --> 00:16:48.020
we never get thank yous.

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You know, the only thing he's that come

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or when maybe when the students are older

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and they realize, you
know, how important and

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what sacrifices teachers made

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and what effort they brought
forth to the classroom.

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So, yeah and I'm sure
it meant a lot to her.

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- Thank you so much Crystal
for that amazing share.

00:17:10.310 --> 00:17:11.660
Let's go over to Theresa.

00:17:11.660 --> 00:17:14.050
Theresa, I'm gonna unmute your line,

00:17:14.050 --> 00:17:15.786
are you still there?

00:17:15.786 --> 00:17:17.240
- [Theresa] Yes.

00:17:17.240 --> 00:17:20.630
So I had a college professor in Algebra

00:17:20.630 --> 00:17:23.130
that was a very difficult class.

00:17:23.130 --> 00:17:26.210
It was my third algebra class
trying to take just to get

00:17:26.210 --> 00:17:28.860
through so I could even take
a college Algebra course

00:17:30.050 --> 00:17:33.400
was feeling very overwhelmed
and quite frankly

00:17:33.400 --> 00:17:35.853
very stupid when it came to Algebra.

00:17:36.760 --> 00:17:41.760
And he told me that even
if you don't get it one way

00:17:43.817 --> 00:17:45.920
and showed me that if
you don't get it one way

00:17:45.920 --> 00:17:48.410
there's always another
way to solve a problem

00:17:49.530 --> 00:17:51.100
which is something that I've taken,

00:17:51.100 --> 00:17:52.650
not just when it comes to math

00:17:52.650 --> 00:17:54.800
but I've taken it into life as well.

00:17:54.800 --> 00:17:57.310
That there's multiple
ways to solve our problems

00:17:58.500 --> 00:18:01.230
and don't just stick
with that one message.

00:18:01.230 --> 00:18:03.310
That's something that I brought
into my teaching as well

00:18:03.310 --> 00:18:05.760
because I do the same
thing with my students now,

00:18:07.050 --> 00:18:08.250
if they can't get it one way

00:18:08.250 --> 00:18:10.670
then let's look at it a different way.

00:18:10.670 --> 00:18:12.440
And it's been very successful

00:18:12.440 --> 00:18:15.070
and has been a good thing for me to learn.

00:18:15.070 --> 00:18:17.720
So I'm very thankful to Mr. Cherry,

00:18:17.720 --> 00:18:19.853
who was an algebra teacher in college.

00:18:20.770 --> 00:18:23.420
- Wow, that's a beautiful story

00:18:23.420 --> 00:18:25.350
and the fact that he
was able to connect it

00:18:25.350 --> 00:18:27.000
or you are connecting it to life

00:18:27.000 --> 00:18:31.710
just that when we hit a
roadblock we have this capacity

00:18:31.710 --> 00:18:34.330
to come up with creative solutions

00:18:34.330 --> 00:18:37.400
or ways of looking at problems differently

00:18:37.400 --> 00:18:40.100
and not just by ourselves,
but even asking for help

00:18:40.100 --> 00:18:41.323
and being able to ask our teachers

00:18:41.323 --> 00:18:46.173
to help us look at something
a new way, that's beautiful.

00:18:47.360 --> 00:18:49.070
- Thank you so much.

00:18:49.070 --> 00:18:50.170
- [Theresa] Thank you.

00:18:51.840 --> 00:18:54.173
- So I would love for us to,

00:18:55.266 --> 00:18:57.010
you might not know this practice

00:18:57.010 --> 00:18:59.940
but it's called loving
kindness meditation,

00:18:59.940 --> 00:19:04.720
and for the skeptics out there,
it can sound kind of wooey

00:19:04.720 --> 00:19:09.170
because we're sending kind
wishes to someone else.

00:19:09.170 --> 00:19:11.360
And it's not that we're in any kind

00:19:11.360 --> 00:19:15.360
of delusional state that
we think we can send wishes

00:19:15.360 --> 00:19:16.960
and they'll somehow be received,

00:19:18.065 --> 00:19:19.300
we're doing it for ourselves.

00:19:19.300 --> 00:19:23.720
There's been a lot of research
on this form of meditation

00:19:23.720 --> 00:19:27.413
and the research has shown
that it connects us to,

00:19:28.290 --> 00:19:29.880
well it has a lot of health benefits,

00:19:29.880 --> 00:19:32.227
reduces inflammation, increases

00:19:32.227 --> 00:19:34.390
and boosts our immune system.

00:19:34.390 --> 00:19:38.500
But it also puts us in touch
with the neurocircuitry

00:19:38.500 --> 00:19:40.370
that makes us wanna connect,

00:19:40.370 --> 00:19:42.300
that helps us attune to each other

00:19:42.300 --> 00:19:45.810
and helps us to be present
for another human being

00:19:45.810 --> 00:19:48.232
and to be present for ourselves.

00:19:48.232 --> 00:19:51.560
So it's really a selfish exercise

00:19:51.560 --> 00:19:53.320
even though we're doing something

00:19:54.590 --> 00:19:57.170
that sounds really benevolent for others

00:19:57.170 --> 00:19:58.790
but it also makes you want,

00:19:58.790 --> 00:20:01.320
sometimes it stimulates this desire

00:20:01.320 --> 00:20:04.250
to reach out to the person or to write

00:20:04.250 --> 00:20:07.170
a letter of thank you or to really just

00:20:07.170 --> 00:20:09.430
be with them in a deeper way.

00:20:09.430 --> 00:20:11.560
So we don't know exactly
what it's gonna do

00:20:11.560 --> 00:20:15.053
but we go into it knowing it
will be in some way beneficial.

00:20:16.240 --> 00:20:19.360
So I would love for us
to send some kind wishes

00:20:19.360 --> 00:20:22.520
to the teachers that you thought of,

00:20:22.520 --> 00:20:26.060
the teacher that came to mind
and I'll lead you through.

00:20:26.060 --> 00:20:28.610
It's a very brief practice,
I'll lead you through it.

00:20:28.610 --> 00:20:30.590
And just keep this person in mind

00:20:30.590 --> 00:20:33.264
and keep the feeling alive in you,

00:20:33.264 --> 00:20:37.490
the feeling of connection of being seen of

00:20:39.127 --> 00:20:44.127
that freedom that you had when you

00:20:44.410 --> 00:20:46.110
from your algebra
teacher, when you learned

00:20:46.110 --> 00:20:47.840
that you could look at
something a different way

00:20:47.840 --> 00:20:49.730
or that there were
multiple ways of solving it

00:20:49.730 --> 00:20:54.700
how that kind of opened
it, like on tied the knot

00:20:54.700 --> 00:20:57.370
that you may have felt when
you didn't get something.

00:20:57.370 --> 00:20:59.220
So just keeping that feeling alive

00:20:59.220 --> 00:21:01.570
and we can close our eyes,

00:21:01.570 --> 00:21:04.090
if we're comfortable, you
can always rest your eyes

00:21:04.090 --> 00:21:05.230
on a spot in front of you,

00:21:05.230 --> 00:21:07.080
if you don't like to close your eyes.

00:21:08.920 --> 00:21:11.363
And let's just bring that teacher to mind.

00:21:15.120 --> 00:21:18.207
You can imagine that they're
in front of you or next to you,

00:21:19.640 --> 00:21:24.200
maybe recall what they
look like, their demeanor

00:21:33.110 --> 00:21:36.270
and just recall the moment of connection

00:21:36.270 --> 00:21:38.113
that you had with this teacher.

00:21:39.840 --> 00:21:42.130
And if no teacher came to
mind, you can just think

00:21:42.130 --> 00:21:44.943
of someone you like or
love, someone in your life.

00:21:52.890 --> 00:21:56.200
And if you'd like, you can
put a hand on your heart

00:21:56.200 --> 00:21:58.073
but you definitely don't have to.

00:21:59.550 --> 00:22:02.440
And I'm just gonna say some
words of kindness out loud

00:22:02.440 --> 00:22:05.270
and just imagine that you're
sending this kindness,

00:22:05.270 --> 00:22:08.943
you're genuinely wishing for
this person to feel this way.

00:22:11.690 --> 00:22:13.083
May you be happy,

00:22:17.010 --> 00:22:19.183
may you know you're appreciated,

00:22:22.930 --> 00:22:26.660
may you be free from
pain, may you be healthy.

00:22:34.290 --> 00:22:38.010
I'm just taking a breath to
reconnect to that feeling

00:22:38.010 --> 00:22:40.933
of appreciation you have for this person.

00:22:43.930 --> 00:22:45.283
May you be happy,

00:22:47.980 --> 00:22:49.373
may you be safe,

00:22:52.040 --> 00:22:54.423
may you be healthy and at ease.

00:22:57.056 --> 00:22:59.223
(sighing)

00:23:04.560 --> 00:23:05.750
And while we're here

00:23:05.750 --> 00:23:09.020
just bringing to mind your own efforts

00:23:09.020 --> 00:23:11.570
you can place your hand
down if it's getting tired.

00:23:12.590 --> 00:23:16.150
But bringing to mind your efforts to show

00:23:16.150 --> 00:23:18.763
up when you don't feel like it sometimes,

00:23:20.540 --> 00:23:23.040
your efforts to be on time,

00:23:23.040 --> 00:23:27.110
to be present your genuine
desire to show care

00:23:31.290 --> 00:23:36.060
and concern for your
students, for your family

00:23:39.060 --> 00:23:43.393
all of that effort and goodness
that you show to the world.

00:23:51.380 --> 00:23:53.020
And if exceptions come up,

00:23:53.020 --> 00:23:54.250
if you start thinking about,

00:23:54.250 --> 00:23:57.380
well not always like
that, that's okay too.

00:23:57.380 --> 00:23:59.660
Just notice what the mind does,

00:23:59.660 --> 00:24:01.940
what the mind shows you greet it with,

00:24:01.940 --> 00:24:05.640
remember that holding with
compassion and kindness

00:24:08.840 --> 00:24:12.150
and we're just gonna send
some kind wishes to ourselves.

00:24:12.150 --> 00:24:13.780
We're gonna use the word you,

00:24:13.780 --> 00:24:17.080
just so we have that little bit of space

00:24:17.080 --> 00:24:22.080
where our loving kind self
wishing it to our vulnerable,

00:24:22.860 --> 00:24:24.823
maybe sometimes hesitant self.

00:24:27.190 --> 00:24:30.420
So may you be happy, may you be safe,

00:24:39.170 --> 00:24:41.343
may you be free from pain,

00:24:44.870 --> 00:24:46.623
may you have ease.

00:24:52.056 --> 00:24:54.620
(inhales) Taking a deep
breath in (exhales)

00:24:54.620 --> 00:24:57.800
and out just to reconnect
with your efforts

00:24:59.065 --> 00:25:00.103
and your goodness.

00:25:02.370 --> 00:25:07.370
May you be happy, may you be safe,

00:25:11.380 --> 00:25:13.553
may you know you're appreciated,

00:25:17.610 --> 00:25:19.613
I may you feel at ease.

00:25:23.890 --> 00:25:27.630
I'm just staying with the feeling of ease

00:25:27.630 --> 00:25:32.113
that maybe has bubbled
up from this exercise,

00:25:33.020 --> 00:25:36.833
we're holding whatever's here, skepticism.

00:25:38.480 --> 00:25:41.250
May be you feeling of
difficulty or hesitation,

00:25:41.250 --> 00:25:44.707
holding that with kindness,
not resisting it (sighs).

00:25:49.125 --> 00:25:54.125
And when you feel ready,
you can open your eyes

00:25:57.210 --> 00:26:00.490
and I just wanna open it up
again to the group to see

00:26:00.490 --> 00:26:04.350
if anyone would like to share
what that was like for you.

00:26:04.350 --> 00:26:06.880
Often, when we do this
loving kindness practice

00:26:06.880 --> 00:26:10.030
for the first time, it can be,

00:26:10.030 --> 00:26:12.820
we have what we call the
inner heckler, right?

00:26:12.820 --> 00:26:15.310
We have this voice inner
voice kind of heckling us

00:26:15.310 --> 00:26:16.650
and saying, what are you doing?

00:26:16.650 --> 00:26:17.623
What is this?

00:26:18.520 --> 00:26:19.930
But maybe you've tried it before,

00:26:19.930 --> 00:26:21.790
maybe it just felt really natural to you.

00:26:21.790 --> 00:26:25.500
So yeah, just feel free to jump in

00:26:25.500 --> 00:26:27.993
and share what that
experience was like for you.

00:26:31.230 --> 00:26:33.750
And again, we can do
it verbally or in chat.

00:26:33.750 --> 00:26:35.854
I love hearing your voices,

00:26:35.854 --> 00:26:38.970
that's my preference,
but whatever you can do

00:26:43.360 --> 00:26:44.210
- [Jeremy] Christy says,

00:26:44.210 --> 00:26:47.373
I'm always surprised of this
exercise can bring me to tears.

00:26:49.540 --> 00:26:50.990
And Abby said, it made me wanna

00:26:50.990 --> 00:26:53.073
call my guiding teacher and say, thanks.

00:26:55.580 --> 00:26:57.950
And then I know there were a couple

00:26:57.950 --> 00:26:59.250
of folks who'd wanna share online,

00:26:59.250 --> 00:27:01.063
so I'm gonna go with Carol first.

00:27:02.180 --> 00:27:04.310
So Carol, I've got open up your line,

00:27:04.310 --> 00:27:07.690
if you wanna unmute yourself,
feel free to say hi to Sam.

00:27:07.690 --> 00:27:09.110
- [Carol] Hello, can you hear me?

00:27:09.110 --> 00:27:10.843
- Yeah, hi Carol.

00:27:10.843 --> 00:27:12.923
- [Carol] Hi, for me,

00:27:14.780 --> 00:27:18.440
I tell my students to do these things

00:27:18.440 --> 00:27:20.590
and I don't always tell myself to.

00:27:20.590 --> 00:27:25.590
And so to hear you, as
the teacher telling me

00:27:25.830 --> 00:27:30.060
as the student to just send myself love,

00:27:30.060 --> 00:27:34.160
it's that kind of switch that I think

00:27:34.160 --> 00:27:36.043
teachers need more often.

00:27:37.681 --> 00:27:39.480
So thank you.

00:27:39.480 --> 00:27:41.550
- You're welcome, it's so true.

00:27:41.550 --> 00:27:42.780
We get in the habit

00:27:42.780 --> 00:27:46.780
of wanting to give advice, give tools

00:27:46.780 --> 00:27:49.893
and then we're we forget to
do it for ourselves, right?

00:27:51.332 --> 00:27:52.770
And we need teachers, right?

00:27:52.770 --> 00:27:55.570
Just like we never
stopped needing teachers

00:27:55.570 --> 00:27:57.433
and helpers as we get older.

00:28:01.760 --> 00:28:04.050
- [Jeremy] And let's go next
to Elizabeth and Elizabeth

00:28:04.050 --> 00:28:06.053
feel free to share your story with Sam.

00:28:06.930 --> 00:28:08.080
- [Elizabeth] Sure, hi.

00:28:10.079 --> 00:28:11.280
I always think it's awesome because

00:28:11.280 --> 00:28:13.390
I have so many people I wanna thank

00:28:13.390 --> 00:28:15.960
and I'm grateful for and appreciate,

00:28:15.960 --> 00:28:18.810
but I can't remember your name

00:28:18.810 --> 00:28:21.920
but you said it brings you to tears.

00:28:21.920 --> 00:28:25.540
And that's what always
surprises me when I do stuff

00:28:25.540 --> 00:28:30.170
like this, you'd come
in and like, it happens

00:28:30.170 --> 00:28:33.260
and mine is, my brother
passed away at age 48,

00:28:33.260 --> 00:28:34.880
a couple of years ago.

00:28:34.880 --> 00:28:38.400
And we had a teacher we
both had in sixth grade

00:28:38.400 --> 00:28:41.600
and she ended up moving
across the street from my mom.

00:28:41.600 --> 00:28:46.210
And 15 years later, she had a picture

00:28:46.210 --> 00:28:49.113
that my brother had given
her like a school picture.

00:28:50.760 --> 00:28:51.920
And she told the story.

00:28:51.920 --> 00:28:53.350
She said, he asked, well

00:28:53.350 --> 00:28:54.900
do you want a school picture of me?

00:28:54.900 --> 00:28:56.610
And she said, well, of course I do.

00:28:56.610 --> 00:28:59.780
And so the next day he
came in with an eight by 10

00:28:59.780 --> 00:29:03.010
and this is like in
the mid early eighties.

00:29:03.010 --> 00:29:07.030
So she kept it and had it when she moved

00:29:07.030 --> 00:29:09.870
in across the street from
my mom, like 15 years later.

00:29:09.870 --> 00:29:13.800
And so it just was a pretty
amazing story that she

00:29:13.800 --> 00:29:16.480
was able to tell us, you
know, at my brother's funeral.

00:29:16.480 --> 00:29:19.150
And like she made a difference then

00:29:19.150 --> 00:29:21.950
and continue to, you know.

00:29:21.950 --> 00:29:25.633
- Wow, what a special woman.

00:29:27.810 --> 00:29:29.233
Thank you for sharing that.

00:29:33.750 --> 00:29:35.470
- [Jeremy] And Sam, do we
have time for one more share?

00:29:35.470 --> 00:29:37.020
- Yes, we do.

00:29:37.020 --> 00:29:40.960
- [Jeremy] Let's go over to
the Jamie (mumbles) actually.

00:29:40.960 --> 00:29:43.560
So Jamie, I'm gonna unmute your line,

00:29:43.560 --> 00:29:45.793
feel free to open it up yourself.

00:29:49.140 --> 00:29:49.973
Hi, Jamie.

00:29:53.380 --> 00:29:55.530
It looks like you might have
a little trouble with Jamie.

00:29:55.530 --> 00:29:56.680
- Oh, okay.

00:29:56.680 --> 00:29:59.500
- [Jeremy] Let me try with Jennifer here.

00:29:59.500 --> 00:30:00.453
Hold on one second.

00:30:02.700 --> 00:30:03.900
Jennifer, are you there?

00:30:11.098 --> 00:30:13.100
Hi, Jennifer, how are you?

00:30:13.100 --> 00:30:15.040
- [Jennifer] I'm good, how are you?

00:30:15.040 --> 00:30:17.230
- Great, thank you so much for joining?

00:30:17.230 --> 00:30:19.030
- [Jennifer] Yeah, again like everyone

00:30:19.030 --> 00:30:21.210
I'm echoing just the sentiments of tears

00:30:21.210 --> 00:30:22.760
and I was surprised

00:30:22.760 --> 00:30:25.270
with how my other senses really connected

00:30:25.270 --> 00:30:27.570
to when I was thinking about
my fourth grade teacher,

00:30:27.570 --> 00:30:29.820
going back to that, I could

00:30:29.820 --> 00:30:31.090
just like it was in the classroom

00:30:31.090 --> 00:30:33.400
and I could hear his
laugh and see his smile

00:30:33.400 --> 00:30:35.800
and it just felt so good.

00:30:35.800 --> 00:30:37.160
And I made the connection that,

00:30:37.160 --> 00:30:38.340
what made him so special,

00:30:38.340 --> 00:30:40.930
he was always a hundred percent present

00:30:40.930 --> 00:30:42.410
with us as his students.

00:30:42.410 --> 00:30:46.563
And that gift of being
present is not forgotten.

00:30:47.460 --> 00:30:52.460
- Yeah, wow, that's incredible.

00:30:52.640 --> 00:30:56.020
The presence it's love, you know

00:30:56.020 --> 00:30:59.150
it's the ultimate show of care

00:30:59.150 --> 00:31:01.940
when you're present and it's hard,

00:31:01.940 --> 00:31:04.840
it gets harder to develop in these times.

00:31:04.840 --> 00:31:09.840
So when we need this time
to stop and just reflect

00:31:11.000 --> 00:31:14.490
and I think we need people
to bring us into it,

00:31:14.490 --> 00:31:16.650
we need all the tools we can get

00:31:16.650 --> 00:31:20.590
to pull us in to this time,
to connect with each other,

00:31:20.590 --> 00:31:23.500
to commune, to share stories.

00:31:23.500 --> 00:31:26.610
So it's so precious and
when we do it, I don't know.

00:31:26.610 --> 00:31:28.830
It sounds like a lot of
you are expressing this too

00:31:28.830 --> 00:31:31.260
but when we do it, we
remember how good it is,

00:31:31.260 --> 00:31:33.590
how good it feels and we wanna do it again

00:31:33.590 --> 00:31:34.530
and again and again.

00:31:34.530 --> 00:31:38.700
So yeah, it's just
rediscovering the importance

00:31:38.700 --> 00:31:41.070
of community and connection, lovely.

00:31:45.050 --> 00:31:48.100
I wanna take us into our next exercise

00:31:48.100 --> 00:31:51.630
which is all about all awe, A-W-E.

00:31:51.630 --> 00:31:53.760
I feel like we don't
use that word very often

00:31:53.760 --> 00:31:55.780
because these moments of awe

00:31:56.770 --> 00:31:59.123
for most of us happen infrequently.

00:32:00.431 --> 00:32:04.300
A moment of awe to me
and how it's been defined

00:32:04.300 --> 00:32:07.880
by the research is these
moments that leave you

00:32:07.880 --> 00:32:11.430
or inspire this feeling of surprise

00:32:11.430 --> 00:32:14.660
and you kind of feel small

00:32:14.660 --> 00:32:19.365
because the thing feels
so big in a good way.

00:32:19.365 --> 00:32:20.570
You feels you feel expansive,

00:32:20.570 --> 00:32:24.620
you feel inspired, you feel,

00:32:24.620 --> 00:32:27.930
you're completely absorbed in that moment,

00:32:27.930 --> 00:32:29.480
you're not thinking about anything else

00:32:29.480 --> 00:32:32.410
because the thing you're
witnessing is so beautiful

00:32:32.410 --> 00:32:35.690
or so touching or so magnificent, right?

00:32:35.690 --> 00:32:39.650
So these are moments
of awe and reflecting,

00:32:39.650 --> 00:32:41.080
there's a lot of research, and I can share

00:32:41.080 --> 00:32:44.924
with you the research
literature that the authors

00:32:44.924 --> 00:32:46.750
there's a woman actually in Canada

00:32:46.750 --> 00:32:49.570
who does this and these
amazing workshops on awe

00:32:49.570 --> 00:32:53.475
but she's found in the research that awe

00:32:53.475 --> 00:32:56.160
is really good for our mental health,

00:32:56.160 --> 00:32:59.420
it stimulates our vagus nerve which again

00:32:59.420 --> 00:33:01.730
activates our parasympathetic
nervous system

00:33:01.730 --> 00:33:06.730
and helps us to actually feel
like we're suspended in time.

00:33:07.420 --> 00:33:10.420
We lose track of time,
we're in this flow state,

00:33:10.420 --> 00:33:12.010
so it's really good for us.

00:33:12.010 --> 00:33:14.400
So I wanna introduce you to this activity

00:33:14.400 --> 00:33:17.160
so that you can put yourself
in that state of awe

00:33:17.160 --> 00:33:19.160
and then hopefully find a lot

00:33:19.160 --> 00:33:22.606
of extraordinary moments
in the ordinariness

00:33:22.606 --> 00:33:25.600
of life, of day-to-day life.

00:33:25.600 --> 00:33:27.610
So what I'm gonna have you do is

00:33:27.610 --> 00:33:29.920
and you can do this again
with your eyes open or closed.

00:33:29.920 --> 00:33:32.980
I like to shut my eyes just to close out

00:33:32.980 --> 00:33:37.490
all the visual stimuli
but we're going to think

00:33:37.490 --> 00:33:40.063
of a moment of awe that we've had.

00:33:41.150 --> 00:33:46.150
So think of a moment
when you were surprised

00:33:47.680 --> 00:33:52.257
and suspended for a moment
in this feeling of amazement,

00:33:55.470 --> 00:34:00.210
this feeling of connection,
maybe it was a connection

00:34:01.402 --> 00:34:06.170
with a person or a moment of
witnessing a child being born

00:34:06.170 --> 00:34:10.890
or seeing a wedding or someone's,

00:34:10.890 --> 00:34:15.530
some experience where
you felt really present

00:34:15.530 --> 00:34:17.543
and in awe of the beauty.

00:34:22.220 --> 00:34:23.580
I'm gonna give you a few moments

00:34:23.580 --> 00:34:26.800
in silence just to allow this memory

00:34:26.800 --> 00:34:29.643
to bubble up without
pushing it or rushing it.

00:34:39.061 --> 00:34:43.644
And as a moment comes to
mind, see if you can expand it

00:34:44.480 --> 00:34:48.760
by noticing the details,
the sensory details.

00:34:48.760 --> 00:34:51.440
So if there were people involved,

00:34:51.440 --> 00:34:53.280
remembering what people were there

00:34:56.290 --> 00:34:58.040
maybe even as you take a deep breath

00:34:58.040 --> 00:35:01.723
you can remember the smell,
the sense in the air.

00:35:05.730 --> 00:35:08.093
Maybe you can remember the sounds,

00:35:13.110 --> 00:35:17.260
maybe you remember the
temperature, if it was a hot day

00:35:17.260 --> 00:35:20.663
or a cold day, neutral.

00:35:25.470 --> 00:35:30.470
And then remember what you
saw the colors, the shapes,

00:35:32.020 --> 00:35:36.703
the facial expressions, the movements.

00:35:43.940 --> 00:35:46.960
Then we're gonna connect with a movement

00:35:46.960 --> 00:35:49.963
that represents this moment of awe.

00:35:51.680 --> 00:35:55.350
So maybe it's your arms
open, or if you were

00:35:57.808 --> 00:36:01.110
holding something, you might
put your arms in that position.

00:36:01.110 --> 00:36:03.720
So just find a movement
in your body right now

00:36:06.552 --> 00:36:08.323
or a gesture that represents
this moment of awe.

00:36:13.200 --> 00:36:16.520
And just hold that movement or posture

00:36:16.520 --> 00:36:20.940
for few breaths, helping
you connect to that feeling

00:36:23.350 --> 00:36:24.650
to let it sock in (sighs).

00:36:34.540 --> 00:36:37.145
Maybe you notice the thoughts that you had

00:36:37.145 --> 00:36:38.495
at the time or the emotions

00:36:43.800 --> 00:36:46.170
and you can place your hands back down

00:36:49.800 --> 00:36:53.380
and just let that feeling simmer

00:36:53.380 --> 00:36:55.343
for just a few more moments.

00:37:15.370 --> 00:37:17.630
Then when you're ready, you can

00:37:19.641 --> 00:37:21.991
if your eyes are closed,
you can open your eyes

00:37:24.343 --> 00:37:25.293
coming back to the space
that we have together.

00:37:27.150 --> 00:37:31.350
And I wanted to share
when I did this exercise

00:37:31.350 --> 00:37:34.130
for the first time, what
moment of awe came to mind?

00:37:34.130 --> 00:37:39.130
And I think Jeremy has a
photograph to pull up to show you.

00:37:41.580 --> 00:37:45.450
It was, when I was 15

00:37:45.450 --> 00:37:47.530
I went to Europe for the first time

00:37:47.530 --> 00:37:52.203
and I saw the Coliseum, oh
that's a beautiful picture.

00:37:53.140 --> 00:37:54.750
Yeah and I looked at it

00:37:54.750 --> 00:37:56.840
and I heard the tour guide starting

00:37:56.840 --> 00:37:59.579
to talk about the history and

00:37:59.579 --> 00:38:03.030
some major points they
were making stuck out

00:38:03.030 --> 00:38:05.800
like how old it was and the events

00:38:05.800 --> 00:38:08.123
that used to take place inside.

00:38:09.178 --> 00:38:12.633
But what got me was how old it was just,

00:38:12.633 --> 00:38:16.400
how it withstood all this time passing

00:38:16.400 --> 00:38:19.030
and all these different events in history

00:38:19.030 --> 00:38:22.170
and it just stood there and it remained.

00:38:22.170 --> 00:38:24.600
And my gesture was this,

00:38:24.600 --> 00:38:26.300
you can't see my hands in the shot

00:38:27.336 --> 00:38:29.760
but like, wow, Kind of pushed back by it

00:38:29.760 --> 00:38:32.470
by how huge it was and how beautiful.

00:38:32.470 --> 00:38:35.720
And I also just felt so lucky to be there,

00:38:35.720 --> 00:38:39.310
I felt so lucky to
witness this living piece

00:38:39.310 --> 00:38:42.560
of history that I had learned
about and in a textbook

00:38:42.560 --> 00:38:43.860
and now I got to see it.

00:38:43.860 --> 00:38:46.400
So that was my moment of awe

00:38:46.400 --> 00:38:50.070
and I just, I really
did lose track of time,

00:38:50.070 --> 00:38:51.400
I think I was there for hours

00:38:51.400 --> 00:38:53.433
but it felt like only minutes went by.

00:38:54.980 --> 00:38:57.480
So I would love to hear from you again

00:38:57.480 --> 00:39:00.630
just what was your moment of
awe, what came up for you?

00:39:00.630 --> 00:39:05.620
What sensory details did you recall?

00:39:05.620 --> 00:39:09.090
And what did it feel like to just stay in

00:39:09.090 --> 00:39:11.150
that feeling of awe?

00:39:11.150 --> 00:39:14.746
There's a lot of research
showing that it takes 30 seconds

00:39:14.746 --> 00:39:19.130
to a minute for a positive
emotion to really register

00:39:19.130 --> 00:39:22.590
in our system for us to
kind of what they say

00:39:22.590 --> 00:39:24.970
or what they call hard-wiring happiness so

00:39:24.970 --> 00:39:29.050
that our default state becomes more likely

00:39:29.050 --> 00:39:32.350
to look for the good, to look
for what we're grateful for,

00:39:32.350 --> 00:39:34.450
to recall these moments of awe.

00:39:34.450 --> 00:39:36.960
It becomes more of a habit over time,

00:39:36.960 --> 00:39:39.088
if we really let it simmer

00:39:39.088 --> 00:39:41.670
and stay with us for
longer periods of time.

00:39:41.670 --> 00:39:43.750
So even talking about it

00:39:43.750 --> 00:39:46.380
and listening to each other is helping us

00:39:46.380 --> 00:39:50.070
to really develop that
hard wiring for the good

00:39:50.070 --> 00:39:52.380
looking for the good, so
we can be fully present

00:39:52.380 --> 00:39:55.740
with each other as we're
listening to these stories of awe.

00:39:55.740 --> 00:39:57.630
And if you'd like, you
can share your gesture

00:39:57.630 --> 00:40:01.750
and I can try to replicate it
and we can all do it together

00:40:01.750 --> 00:40:03.180
so that we're mirroring each other

00:40:03.180 --> 00:40:05.800
and really maybe even touching deeply

00:40:05.800 --> 00:40:08.123
into empathizing with your experience.

00:40:09.380 --> 00:40:10.933
So who would like to share?

00:40:11.890 --> 00:40:13.430
- [Jeremy] We heard a number of great ones

00:40:13.430 --> 00:40:15.970
in the question section, Sam needs to talk

00:40:15.970 --> 00:40:18.593
about sunset on the rim
of the Grand Canyon.

00:40:19.512 --> 00:40:21.600
Jackline talked about
the first time I was out

00:40:21.600 --> 00:40:24.860
on the ocean without being
able to see the shore.

00:40:24.860 --> 00:40:26.700
You met said the moment of owe

00:40:26.700 --> 00:40:28.480
that really struck her
when she was presented

00:40:28.480 --> 00:40:30.270
with her daughter for the first time

00:40:30.270 --> 00:40:32.483
and she looked in her daughter's eyes

00:40:32.483 --> 00:40:33.970
and has had a connection with her,

00:40:33.970 --> 00:40:35.920
this lasted all the way into adulthood.

00:40:38.800 --> 00:40:42.160
- It's an incredible
moment, I'm not a parent yet

00:40:42.160 --> 00:40:46.470
but I imagine that moment
of, I mean overwhelming awe

00:40:46.470 --> 00:40:49.033
when your child is placed in your arms.

00:40:50.310 --> 00:40:53.500
And sometimes we'll do
these exercises with parents

00:40:53.500 --> 00:40:56.020
and kids and have them
look into each other's eyes

00:40:56.020 --> 00:40:58.230
and talk about the color of their eyes.

00:40:58.230 --> 00:41:00.410
Like I noticed you have a little ring

00:41:00.410 --> 00:41:02.775
around your eye and it's
light brown or whatever it is

00:41:02.775 --> 00:41:06.770
but for me witnessing
those moments of connection

00:41:06.770 --> 00:41:10.730
and seeing parents just
their whole face light up

00:41:10.730 --> 00:41:13.810
as their child, children
are looking into their eyes

00:41:13.810 --> 00:41:16.239
is for me, that's a moment of awe

00:41:16.239 --> 00:41:21.080
that I happily get to
relive a lot of the time.

00:41:21.080 --> 00:41:24.033
So yeah, just thank you for sharing that.

00:41:25.590 --> 00:41:27.603
Let's invite some folks that join live.

00:41:28.690 --> 00:41:30.460
Let's start with Michelle.

00:41:30.460 --> 00:41:32.810
So Michelle, I'm gonna unmute your line,

00:41:32.810 --> 00:41:34.710
feel free to do the same on your side.

00:41:38.860 --> 00:41:40.510
Michelle, are you there?

00:41:40.510 --> 00:41:42.360
- [Michelle] Yes, I am.

00:41:42.360 --> 00:41:43.770
- Hi Michelle.

00:41:43.770 --> 00:41:46.653
- [Michelle] Hi, this is amazing.

00:41:47.990 --> 00:41:51.720
The first time I held each of my children

00:41:51.720 --> 00:41:56.720
and it washed over me the enormity of

00:41:59.040 --> 00:42:01.800
how my life would change after that

00:42:02.860 --> 00:42:06.300
life as I had known, it
would never be the same

00:42:06.300 --> 00:42:11.300
and in ways I couldn't
even at that time imagine,

00:42:14.595 --> 00:42:18.928
so that, that was awesome (laughs).

00:42:24.250 --> 00:42:27.780
- Beautiful, I got chills
all through my head

00:42:27.780 --> 00:42:29.933
and back as you were describing that.

00:42:31.650 --> 00:42:33.300
- [Michelle] Took my breath away.

00:42:38.430 --> 00:42:39.263
- That's beautiful.

00:42:39.263 --> 00:42:40.830
Was there a movement that came to mind?

00:42:40.830 --> 00:42:42.602
Was it like holding?

00:42:42.602 --> 00:42:47.602
- [Michelle] Yes, it was as
I cradled them in my arms

00:42:48.590 --> 00:42:50.963
and looked down at their little faces.

00:42:54.610 --> 00:42:58.040
And that was so many years
ago they're adults now,

00:42:58.040 --> 00:43:03.040
but I can cherish that moment once again,

00:43:04.570 --> 00:43:06.630
so I appreciate this.

00:43:06.630 --> 00:43:10.280
I hadn't thought about
that in many, many years,

00:43:10.280 --> 00:43:12.920
so thank you for that.

00:43:12.920 --> 00:43:15.593
- Yeah, you're so welcome,
thank you for sharing.

00:43:17.100 --> 00:43:22.100
Sometimes I look into, I work
with children quite a bit

00:43:22.300 --> 00:43:25.100
and I look into their eyes
as if I was their parent.

00:43:25.100 --> 00:43:28.130
Like I try to connect with
them with that kind of love

00:43:28.130 --> 00:43:31.850
and unconditional positive regard,

00:43:31.850 --> 00:43:33.610
if you wanna use therapy terms

00:43:33.610 --> 00:43:35.880
but that feeling of connection.

00:43:35.880 --> 00:43:37.870
And we can kind of do that

00:43:37.870 --> 00:43:39.747
with other people, even
strangers, we can look

00:43:39.747 --> 00:43:44.190
at them through the eyes of
a mother or a sister friend

00:43:44.190 --> 00:43:49.190
and really connect to
each other on this level.

00:43:49.770 --> 00:43:51.330
You know, we have that capacity

00:43:51.330 --> 00:43:53.920
reminds me of the algebra teacher,

00:43:53.920 --> 00:43:57.860
kind of asking us to look at
the problem of disconnection

00:43:57.860 --> 00:44:01.273
or loneliness and maybe
one solution is to recall

00:44:03.010 --> 00:44:05.060
that feeling of love that we have

00:44:05.060 --> 00:44:07.380
for people in our lives and to maybe look

00:44:07.380 --> 00:44:10.003
at each other through that lens.

00:44:12.589 --> 00:44:15.033
- [Jeremy] Let's go over to Lindsay.

00:44:15.880 --> 00:44:18.270
Lindsay, I'm gonna unmute your line,

00:44:18.270 --> 00:44:20.080
feel free to connect with Sam.

00:44:20.080 --> 00:44:21.770
- [Lindsay] Hi there,
can you guys hear me?

00:44:21.770 --> 00:44:23.103
- Yeah, hi Lindsey.

00:44:24.400 --> 00:44:28.000
- Thank you so much Sam, I
appreciate this, this is awesome.

00:44:28.000 --> 00:44:29.870
So for me, it was also
when I was traveling

00:44:29.870 --> 00:44:31.180
and I think there was a few of us

00:44:31.180 --> 00:44:34.160
said we were somewhere
far, far away from home.

00:44:34.160 --> 00:44:36.910
So I was in *** me and
(mumbles) a couple of years ago

00:44:36.910 --> 00:44:39.000
and I was watching the sunrise

00:44:39.000 --> 00:44:42.400
over hundreds of thousands of pagodas

00:44:42.400 --> 00:44:45.810
and it literally just
seemed like time stopped

00:44:45.810 --> 00:44:47.890
and the sun took forever to come up,

00:44:47.890 --> 00:44:49.840
the transformation of the sky.

00:44:49.840 --> 00:44:52.990
And so that was really
nice to kind of go back

00:44:52.990 --> 00:44:54.230
and revisit that I haven't looked

00:44:54.230 --> 00:44:55.430
at those photos in forever

00:44:55.430 --> 00:44:58.380
but I was sitting on the
pagoda, looking at the sunrise,

00:44:58.380 --> 00:45:00.990
I was right there back in that moment.

00:45:00.990 --> 00:45:04.410
And I try not to focus
too much on the past,

00:45:04.410 --> 00:45:07.970
try to focus on the present
and maybe the next day

00:45:07.970 --> 00:45:09.660
or the next week but
this was a nice reminder

00:45:09.660 --> 00:45:12.590
that it is okay to kind
of go back to the past

00:45:12.590 --> 00:45:15.010
and think of those, those moments of awe,

00:45:15.010 --> 00:45:17.113
so thank you for that so much.

00:45:18.116 --> 00:45:19.690
- Oh, thank you.

00:45:19.690 --> 00:45:21.430
Again, I got chills for my whole body.

00:45:21.430 --> 00:45:24.240
I could see them as these pagodas

00:45:24.240 --> 00:45:26.190
as you were describing them in my mind.

00:45:27.170 --> 00:45:28.922
You know, when we listened to stories

00:45:28.922 --> 00:45:31.430
each other's stories, we're giving gifts.

00:45:31.430 --> 00:45:33.560
That's how I see stories
where we're really

00:45:33.560 --> 00:45:37.970
gifting each other with
these pleasant experiences

00:45:37.970 --> 00:45:40.340
because our brain doesn't
really know the difference

00:45:40.340 --> 00:45:43.080
between living it, like being there again

00:45:43.080 --> 00:45:45.400
and being here now,

00:45:45.400 --> 00:45:47.980
the only differences are our motor cortex

00:45:47.980 --> 00:45:51.280
which is activated when we're
actually physically moving.

00:45:51.280 --> 00:45:53.900
But our brain, you know,
going back to the past

00:45:53.900 --> 00:45:57.714
and really reliving some of
these moments is so good.

00:45:57.714 --> 00:46:01.880
The researcher who I love and
I've done these workshops with

00:46:01.880 --> 00:46:04.730
she talks about it like
a multivitamin, you know,

00:46:04.730 --> 00:46:06.510
the benefits you're giving yourself

00:46:06.510 --> 00:46:09.040
by recalling the past and sharing stories

00:46:09.040 --> 00:46:13.410
and listening to stories is
medicinal in so many ways.

00:46:13.410 --> 00:46:18.410
So yeah, I encourage you all
to really go there often.

00:46:19.110 --> 00:46:21.670
and every time we go there,
we're reinforcing it,

00:46:21.670 --> 00:46:25.550
so it's easier to drop in, before bed,

00:46:25.550 --> 00:46:27.780
if you're having a hard
time falling asleep,

00:46:27.780 --> 00:46:31.540
go to the pagodas, go to your child,

00:46:31.540 --> 00:46:35.430
looking into their eyes,
go to the Coliseum,

00:46:35.430 --> 00:46:39.870
wherever it is, but touch
into it because it's this pool

00:46:39.870 --> 00:46:44.870
of nourishment and it
really, you know changes.

00:46:46.600 --> 00:46:50.090
I think hopefully you all feel a shift,

00:46:50.090 --> 00:46:51.960
our nervous system has settled down

00:46:51.960 --> 00:46:55.430
and if I was just
listening to these stories

00:46:55.430 --> 00:46:58.040
after a busy day and didn't ground myself

00:46:58.040 --> 00:47:01.090
I wouldn't have gotten all
these wonderful sensations

00:47:01.090 --> 00:47:04.435
of tingling and warmth and
the visions of your stories.

00:47:04.435 --> 00:47:09.435
So we need that rest,
we need that stillness

00:47:09.760 --> 00:47:14.110
and rejuvenation to then access
each other more intimately

00:47:14.110 --> 00:47:17.463
and have empathy this deep,
deep embodied empathy.

00:47:19.110 --> 00:47:21.710
So that's what I would
love you to take away

00:47:21.710 --> 00:47:25.850
the time has flown by, but that it,

00:47:25.850 --> 00:47:30.820
to take away looking for
the good, it's a practice,

00:47:30.820 --> 00:47:34.740
looking around and even just
the space you're in right now

00:47:34.740 --> 00:47:39.630
and appreciating, I have
photos of my family up here

00:47:39.630 --> 00:47:41.010
and all these wonderful memories

00:47:41.010 --> 00:47:44.283
but just looking for the good and finding

00:47:44.283 --> 00:47:47.449
extraordinariness in the ordinary

00:47:47.449 --> 00:47:50.590
especially in your students,
and maybe even try looking

00:47:50.590 --> 00:47:55.120
at them through the eyes of a parent

00:47:55.120 --> 00:48:00.110
or that feeling of unconditional
positive regard, even,

00:48:00.110 --> 00:48:02.490
especially I should say
the most difficult students

00:48:02.490 --> 00:48:06.910
who challenge us in so many ways, but yeah

00:48:06.910 --> 00:48:11.690
also teach her some
really beautiful lessons.

00:48:11.690 --> 00:48:15.250
So it has just been such
a pleasure to be with you

00:48:15.250 --> 00:48:19.630
I really hope that you
feel refreshed, rejuvenated

00:48:19.630 --> 00:48:22.880
that the tank is a little bit more full

00:48:22.880 --> 00:48:25.709
and we can just close our
eyes for one last moment

00:48:25.709 --> 00:48:30.709
and just connect there, how
many people on this cal?

00:48:30.720 --> 00:48:35.720
Over 400, 600, 650
people that we can't see,

00:48:35.749 --> 00:48:39.750
we heard some voices, but
we know they're there.

00:48:39.750 --> 00:48:44.750
All of us educator, teaching,
learning from our students

00:48:45.030 --> 00:48:48.690
giving of ourselves, we're
part of this community

00:48:49.610 --> 00:48:52.000
even though we can't see
each other, we're here

00:48:54.797 --> 00:48:58.710
to just taking a moment
to send these kind wishes

00:48:58.710 --> 00:49:02.140
to our whole community,
can think of these beams

00:49:02.140 --> 00:49:05.030
of light going out from
our home all the way around

00:49:05.030 --> 00:49:08.300
the world and at least the country,

00:49:08.300 --> 00:49:11.513
but we can include the
world community of teachers.

00:49:13.290 --> 00:49:18.290
May we all find joy, may
we all have as many moments

00:49:23.210 --> 00:49:26.303
of awe as possible in our lives.

00:49:30.080 --> 00:49:32.283
May we all know we're not alone,

00:49:35.720 --> 00:49:39.913
may we all feel connected
and safe and healthy.

00:49:43.124 --> 00:49:46.050
(exhales)

00:49:46.050 --> 00:49:47.290
And you've sent that out

00:49:47.290 --> 00:49:50.200
and now just take a few breaths receiving

00:49:50.200 --> 00:49:53.720
from the 650 people
that have sent it to you

00:49:53.720 --> 00:49:58.720
just breathing in all the
good that's been sent your way

00:50:00.340 --> 00:50:04.683
and letting it land in your
body, your mind, your heart.

00:50:14.510 --> 00:50:18.400
Letting this feeling of
connection and ease, linger

00:50:23.070 --> 00:50:26.173
and just taking it into
the rest of your evening.

00:50:27.500 --> 00:50:31.140
Maybe the next person you make eye contact

00:50:31.140 --> 00:50:35.300
with just sending them
kindness through your eyes

00:50:41.070 --> 00:50:45.870
and can open our eyes and
for all the timekeepers,

00:50:45.870 --> 00:50:47.900
it's exactly five o'clock.

00:50:47.900 --> 00:50:50.570
So I wanna honor and respect your time

00:50:51.450 --> 00:50:55.690
and just say thank you so
much to our whole team,

00:50:55.690 --> 00:50:57.840
to Jeremy and Alice and Leno

00:50:57.840 --> 00:51:01.950
for putting this all together
and all of you for coming.

00:51:01.950 --> 00:51:05.210
It's been a pleasure and I
hope we can do this again soon.

00:51:05.210 --> 00:51:06.463
Really genuinely.

00:51:11.580 --> 00:51:13.160
- Thank you so much, Sam.

00:51:13.160 --> 00:51:15.825
I just wanted to end on
this note from Michelle

00:51:15.825 --> 00:51:16.790
which I thought was so perfect.

00:51:16.790 --> 00:51:19.400
Michelle says thank you
for this brief respite

00:51:19.400 --> 00:51:20.517
from the rigors of the day to day.

00:51:20.517 --> 00:51:23.080
And we all know that
teaching has those rigors,

00:51:23.080 --> 00:51:24.520
it was a precious gift.

00:51:24.520 --> 00:51:27.530
Maybe you also find joy
and time that you can share

00:51:27.530 --> 00:51:29.470
with others in this wonderful practice.

00:51:29.470 --> 00:51:31.370
So thank you all for joining today.

00:51:31.370 --> 00:51:34.110
Thank you, Sam, for
sharing your gifts with us

00:51:34.110 --> 00:51:36.180
and we wish you all a wonderful spring.

00:51:36.180 --> 00:51:37.013
- Thank you.

00:51:38.240 --> 00:51:39.163
Bye everybody.


Subtracting vectors with parallelogram rule

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.430
- [Instructor] In this
video, we're gonna think

00:00:01.430 --> 00:00:04.140
about what it means to subtract vectors,

00:00:04.140 --> 00:00:06.620
especially in the context
of what we talked about

00:00:06.620 --> 00:00:08.760
as the parallelogram rule.

00:00:08.760 --> 00:00:11.880
So let's say we want
to start with vector a

00:00:11.880 --> 00:00:15.360
and from that we want
to subtract vector b.

00:00:15.360 --> 00:00:18.300
And we have vectors a and b depicted here.

00:00:18.300 --> 00:00:19.740
What do you think this is going to be?

00:00:19.740 --> 00:00:22.210
What do you think is going
to be the resulting vector?

00:00:22.210 --> 00:00:24.110
Pause this video and think about that.

00:00:25.060 --> 00:00:25.893
All right.

00:00:25.893 --> 00:00:30.893
Now the key thing to realize
is a minus b is the same thing

00:00:31.360 --> 00:00:36.360
as vector a plus the negative of vector b.

00:00:38.720 --> 00:00:42.000
Now, what is the negative
of vector b look like?

00:00:42.000 --> 00:00:43.850
Well, that's going to be a vector

00:00:43.850 --> 00:00:46.980
that has the exact same
magnitude as vector b

00:00:46.980 --> 00:00:49.500
but just in the opposite direction.

00:00:49.500 --> 00:00:53.320
For example this vector right over here

00:00:53.320 --> 00:00:55.043
would be the vector -b.

00:00:56.501 --> 00:00:57.960
Now we just have to think about

00:00:57.960 --> 00:01:01.230
what is vector a plus the vector -b?

00:01:01.230 --> 00:01:03.400
Well, there's two ways
of thinking about that.

00:01:03.400 --> 00:01:05.200
I could put the tails of both of them

00:01:05.200 --> 00:01:08.010
at the same starting point,
might as well do the origin.

00:01:08.010 --> 00:01:10.650
So let me draw -b over here.

00:01:10.650 --> 00:01:15.650
So we know the vector -b looks like that.

00:01:16.160 --> 00:01:18.410
So one way that you are probably familiar

00:01:18.410 --> 00:01:22.630
is you have vector a and then
what you do is you take a copy

00:01:22.630 --> 00:01:24.690
or you could think of shifting vector b

00:01:24.690 --> 00:01:27.940
so its tail starts at
the head of vector a.

00:01:27.940 --> 00:01:32.940
And if you did that, it
would look like this.

00:01:33.100 --> 00:01:35.060
It would look like this.

00:01:35.060 --> 00:01:36.553
This is also the vector -b.

00:01:38.060 --> 00:01:42.553
And then the sum of vector a and vector -b

00:01:44.210 --> 00:01:48.570
is going to be going
from the tail of vector a

00:01:48.570 --> 00:01:50.802
to the head of vector -b.

00:01:50.802 --> 00:01:55.510
So this would be the
result, right over here.

00:01:55.510 --> 00:01:58.860
Which you could view
as the sum of a plus -b

00:01:58.860 --> 00:02:01.650
or the difference of vectors a and b

00:02:01.650 --> 00:02:04.070
or vector a minus vector b.

00:02:04.070 --> 00:02:05.230
Now, if we wanna think about it

00:02:05.230 --> 00:02:07.400
in terms of the parallelogram rule,

00:02:07.400 --> 00:02:09.504
we could take another copy of vector a

00:02:09.504 --> 00:02:14.380
and put it so that it's
tail's at the head of this -b

00:02:14.380 --> 00:02:18.220
and then we would get it right over here

00:02:18.220 --> 00:02:20.400
and we are forming the parallelogram.

00:02:20.400 --> 00:02:22.340
And then the resulting vector

00:02:22.340 --> 00:02:24.960
is the diagonal of the parallelogram.

00:02:24.960 --> 00:02:26.900
And this just helps us appreciate

00:02:26.900 --> 00:02:28.510
that we could start with -b

00:02:28.510 --> 00:02:31.540
and then add vector a to that.

00:02:31.540 --> 00:02:33.510
Or we could start with vector a

00:02:33.510 --> 00:02:35.810
and then add -b to that.

00:02:35.810 --> 00:02:39.370
But either way you get this
white vector right over here

00:02:39.370 --> 00:02:44.370
which we can view as the
vector a minus vector b,

00:02:44.620 --> 00:02:45.453
and we're done.


Adding vectors in magnitude and direction form

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.030
- [Instructor] We're told that
vector a has magnitude four

00:00:03.030 --> 00:00:06.340
and direction 170 degrees
from the positive x-axis.

00:00:06.340 --> 00:00:09.180
Vector b has magnitude three
and direction 240 degrees

00:00:09.180 --> 00:00:10.960
from the positive x-axis.

00:00:10.960 --> 00:00:15.490
Find the magnitude and direction
of vector a plus vector b.

00:00:15.490 --> 00:00:18.510
So pause this video and see
if you can have a go at that.

00:00:18.510 --> 00:00:19.960
All right, now let's work
through this together.

00:00:19.960 --> 00:00:21.390
And the way that I'm going to approach it,

00:00:21.390 --> 00:00:24.410
I'm going to represent each
vector in component form.

00:00:24.410 --> 00:00:26.270
And then I'm going to add
the corresponding components.

00:00:26.270 --> 00:00:28.650
And from that, I'll try to
figure out the magnitude

00:00:28.650 --> 00:00:30.550
and the direction of the sum.

00:00:30.550 --> 00:00:33.660
So vector a, what is its x-component?

00:00:33.660 --> 00:00:36.160
Well, the change in x
here, there's multiple ways

00:00:36.160 --> 00:00:39.150
that you could try to do
this using trigonometry.

00:00:39.150 --> 00:00:42.390
But we've reviewed this or
gone over this in other videos.

00:00:42.390 --> 00:00:43.830
The simplest way to think about it is

00:00:43.830 --> 00:00:46.057
our change in x here is
going to be the length.

00:00:46.057 --> 00:00:49.620
And we know vector a has
magnitude four times the cosine

00:00:49.620 --> 00:00:51.160
of the angle that the vector makes

00:00:51.160 --> 00:00:54.910
with the positive x-axis,
cosine of 170 degrees.

00:00:54.910 --> 00:00:56.730
And so that's our
x-component right over here,

00:00:56.730 --> 00:01:00.530
four times cosine of 170 degrees.

00:01:00.530 --> 00:01:03.140
And then what's our y-component?

00:01:03.140 --> 00:01:06.670
Well, our y-component is going
to be this change in y here.

00:01:06.670 --> 00:01:08.870
And as we've reviewed in other videos,

00:01:08.870 --> 00:01:11.470
that's going to be the
length times the sine

00:01:11.470 --> 00:01:13.810
of the angle we make
with a positive x-axis,

00:01:13.810 --> 00:01:17.800
sine of 170 degrees.

00:01:17.800 --> 00:01:19.760
And we can maybe use a calculator in a bit

00:01:19.760 --> 00:01:22.130
to get approximations for these values.

00:01:22.130 --> 00:01:25.750
But then we can do the exact
same thing for vector b.

00:01:25.750 --> 00:01:30.670
Vector b here is going
to be, by the same logic,

00:01:30.670 --> 00:01:34.290
it's x-component is going to
be the length of the vector,

00:01:34.290 --> 00:01:36.230
and it would be three.

00:01:36.230 --> 00:01:37.220
They tell us that.

00:01:37.220 --> 00:01:39.930
So it's going to be three times the cosine

00:01:39.930 --> 00:01:43.290
of this angle, 240 degrees.

00:01:43.290 --> 00:01:46.990
And then the y-component
is going to be the length

00:01:46.990 --> 00:01:50.200
of our vector three times the sine

00:01:50.200 --> 00:01:53.650
of 240 degrees.

00:01:53.650 --> 00:01:56.270
Now, when we wanna take
the sum of the two vectors,

00:01:56.270 --> 00:01:57.103
let me write it here,

00:01:57.103 --> 00:02:01.350
vector a plus vector b,

00:02:01.350 --> 00:02:04.490
I can just add the
corresponding components.

00:02:04.490 --> 00:02:08.520
This is going to be equal to four cosine

00:02:08.520 --> 00:02:13.520
of 170 degrees plus three cosine

00:02:14.130 --> 00:02:16.160
of 240 degrees.

00:02:16.160 --> 00:02:20.137
And then the y-component
is going to be four sine

00:02:21.880 --> 00:02:26.880
of 170 degrees plus three sine

00:02:27.860 --> 00:02:30.750
of 240 degrees.

00:02:30.750 --> 00:02:34.360
And so let me get my calculator
out to evaluate these.

00:02:34.360 --> 00:02:36.350
We say 170 degrees.

00:02:36.350 --> 00:02:41.350
We take the cosine times
four, that equals this.

00:02:41.720 --> 00:02:43.500
And then we're going to add to that.

00:02:43.500 --> 00:02:45.430
I'll open parentheses.

00:02:45.430 --> 00:02:47.240
We'll take the cosine of 240.

00:02:47.240 --> 00:02:52.240
240 cosine times three, close parentheses,

00:02:53.000 --> 00:02:54.840
is equal to this, negative,

00:02:54.840 --> 00:02:58.210
approximately negative 5.44.

00:02:58.210 --> 00:03:03.210
So this is approximately negative 5.44.

00:03:03.690 --> 00:03:07.120
And then if we were to take 170 degrees,

00:03:07.120 --> 00:03:10.740
take the sine of it, multiply it by four.

00:03:10.740 --> 00:03:13.670
And then to that, I'm
going to open parentheses.

00:03:13.670 --> 00:03:17.310
I'm gonna take 240 degrees, take the sine,

00:03:17.310 --> 00:03:20.950
multiply that times three,
close my parentheses.

00:03:20.950 --> 00:03:23.800
That is going to be equal to
approximately negative 1.90.

00:03:25.250 --> 00:03:28.773
So this is approximately negative 1.90.

00:03:30.280 --> 00:03:32.460
And this is consistent with our intuition.

00:03:32.460 --> 00:03:35.590
If the sum has both negative components,

00:03:35.590 --> 00:03:37.240
that means it's going to
be in the third quadrant.

00:03:37.240 --> 00:03:39.000
And if I were to do
the head to tail method

00:03:39.000 --> 00:03:41.120
of adding vectors, if
I were to take vector b

00:03:41.120 --> 00:03:44.080
and I were to put it right over here,

00:03:44.080 --> 00:03:45.890
we see that the resulting vector,

00:03:45.890 --> 00:03:48.960
the sum will sit in the third quadrant.

00:03:48.960 --> 00:03:51.540
It makes sense that our x and y-components

00:03:51.540 --> 00:03:53.120
would indeed be negative.

00:03:53.120 --> 00:03:54.270
Now, the question didn't ask

00:03:54.270 --> 00:03:56.070
just to find the components of the sum.

00:03:56.070 --> 00:03:59.210
It asked to find the
magnitude and the direction

00:03:59.210 --> 00:04:00.980
of the resulting sum.

00:04:00.980 --> 00:04:02.320
And so to do that,

00:04:02.320 --> 00:04:05.730
we just have to use a little
bit more of our trigonometry

00:04:05.730 --> 00:04:07.880
and actually a little bit of our geometry.

00:04:07.880 --> 00:04:12.210
For example, our change in x
is this value right over here

00:04:12.210 --> 00:04:14.350
as we go from the tail to the tip.

00:04:14.350 --> 00:04:16.650
It's negative 5.44.

00:04:16.650 --> 00:04:19.260
If we're just thinking in terms
of length right over here,

00:04:19.260 --> 00:04:23.563
the absolute value, this
side would have length 5.44.

00:04:24.880 --> 00:04:27.040
And then same way, you are
changing y, its negative,

00:04:27.040 --> 00:04:29.080
we're going down in y.

00:04:29.080 --> 00:04:30.992
But if we were just thinking
in terms of a triangle,

00:04:30.992 --> 00:04:33.533
the length on this side
of a triangle is 1.90.

00:04:35.570 --> 00:04:38.500
And we can see from
the Pythagorean theorem

00:04:38.500 --> 00:04:42.400
that the length of our hypotenuse,

00:04:42.400 --> 00:04:44.670
which is the same thing as
the magnitude of this vector,

00:04:44.670 --> 00:04:47.400
squared is going to be equal
to the sum of the squares

00:04:47.400 --> 00:04:48.710
of these two sides.

00:04:48.710 --> 00:04:50.280
Or another way of thinking about it is,

00:04:50.280 --> 00:04:53.560
the length of this vector,
the magnitude of this vector,

00:04:53.560 --> 00:04:56.140
which we can write as
a magnitude of vector a

00:04:56.140 --> 00:05:00.070
plus vector b is going to be equal to,

00:05:00.070 --> 00:05:01.660
or I should say approximately equal to

00:05:01.660 --> 00:05:04.350
since we're already
approximating these values,

00:05:04.350 --> 00:05:09.350
the principal root of 5.44 squared.

00:05:09.620 --> 00:05:10.660
And that's 'cause I'm just thinking

00:05:10.660 --> 00:05:12.750
about the absolute length of the side.

00:05:12.750 --> 00:05:14.330
I could also think about a change in x.

00:05:14.330 --> 00:05:16.770
But if I had a negative
5.44 and I square that,

00:05:16.770 --> 00:05:18.440
that would still become positive.

00:05:18.440 --> 00:05:23.000
And then I'll have plus 1.90 squared.

00:05:23.000 --> 00:05:25.330
And I can get our calculator out for that.

00:05:25.330 --> 00:05:26.860
This is going to be

00:05:26.860 --> 00:05:31.860
approximately equal to 5.44 squared

00:05:32.490 --> 00:05:37.000
plus 1.9 squared,

00:05:37.000 --> 00:05:38.330
is equal to that.

00:05:38.330 --> 00:05:40.120
Take the square root of that.

00:05:40.120 --> 00:05:43.533
It's approximately equal to 5.76,

00:05:45.540 --> 00:05:50.010
5.76, which is going to be our magnitude.

00:05:50.010 --> 00:05:51.880
And then to figure out the direction,

00:05:51.880 --> 00:05:54.610
so we essentially want to figure out

00:05:54.610 --> 00:05:57.210
this angle right over here.

00:05:57.210 --> 00:06:01.340
You might recognize that
the tangent of this angle,

00:06:01.340 --> 00:06:02.720
theta right over here,

00:06:02.720 --> 00:06:05.790
should be equal to, and I'll
do approximately equal to

00:06:05.790 --> 00:06:07.810
since we're using these approximations,

00:06:07.810 --> 00:06:11.280
our change in y over our change in x.

00:06:11.280 --> 00:06:12.190
So negative 1.90

00:06:14.020 --> 00:06:17.920
over negative 5.44,

00:06:17.920 --> 00:06:20.070
or we could say that theta

00:06:20.070 --> 00:06:24.640
is going to be approximately
equal to the inverse tangent

00:06:24.640 --> 00:06:26.150
of negative 1.90

00:06:29.100 --> 00:06:33.980
over negative 5.44.

00:06:33.980 --> 00:06:35.500
And we're gonna see in a second

00:06:35.500 --> 00:06:37.560
whether this is actually
going to get us the answer

00:06:37.560 --> 00:06:38.393
that we want.

00:06:38.393 --> 00:06:39.890
So let's try this out.

00:06:39.890 --> 00:06:44.210
If we were to take 1.9 negative

00:06:44.210 --> 00:06:48.270
divided by 5.44 negative,

00:06:48.270 --> 00:06:49.770
that gets us that, which makes sense.

00:06:49.770 --> 00:06:52.100
Negative divided by a
negative is a positive.

00:06:52.100 --> 00:06:55.490
And now let's try to take
the inverse tangent of that.

00:06:55.490 --> 00:06:59.290
So here I press second, and
then I'll do inverse tangent.

00:06:59.290 --> 00:07:00.123
So I'm getting

00:07:00.123 --> 00:07:04.740
19.25 degrees, approximately.

00:07:04.740 --> 00:07:05.650
So this is saying

00:07:05.650 --> 00:07:10.420
that this is approximately 19.25 degrees.

00:07:10.420 --> 00:07:13.263
And my question to you
is, does that seem right?

00:07:14.490 --> 00:07:18.030
Well, 19.25 degrees would
put us in the first quadrant.

00:07:18.030 --> 00:07:19.070
It would get us a vector

00:07:19.070 --> 00:07:22.630
that looks something like this.

00:07:22.630 --> 00:07:26.630
This would be 19.25 degrees.

00:07:26.630 --> 00:07:28.680
But clearly, that's not the
vector we're talking about.

00:07:28.680 --> 00:07:31.120
We're talking about a vector
in the third quadrant.

00:07:31.120 --> 00:07:33.190
And the reason why we got this result,

00:07:33.190 --> 00:07:34.900
is that when you take the inverse tangent

00:07:34.900 --> 00:07:36.290
on most calculators,

00:07:36.290 --> 00:07:37.920
it's going to give you an angle

00:07:37.920 --> 00:07:39.840
that's between negative 90 degrees

00:07:39.840 --> 00:07:41.690
and positive 90 degrees.

00:07:41.690 --> 00:07:43.300
While here we are at an angle

00:07:43.300 --> 00:07:45.370
that puts us out in the third quadrant.

00:07:45.370 --> 00:07:46.870
So we have to adjust.

00:07:46.870 --> 00:07:51.060
And to adjust here, we just
have to add 180 degrees

00:07:51.060 --> 00:07:54.580
to get to the actual angle
that we are talking about.

00:07:54.580 --> 00:07:57.920
So in our situation, the magnitude here

00:07:57.920 --> 00:08:01.200
is going to be approximately 5.76

00:08:03.690 --> 00:08:07.010
and then the direction is going to be

00:08:07.010 --> 00:08:12.010
approximately 19.25 plus 180 degrees,

00:08:12.100 --> 00:08:14.970
which is going to be 199.25 degrees.

00:08:17.820 --> 00:08:19.423
And now we are done.


Parallelogram rule for vector addition

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:01.460
- [Instructor] So we
have two vectors here,

00:00:01.460 --> 00:00:03.280
vector A and vector B.

00:00:03.280 --> 00:00:05.110
And what we're gonna do in this video

00:00:05.110 --> 00:00:08.480
is think about what it
means to add vectors.

00:00:08.480 --> 00:00:10.180
So for example, how could we think

00:00:10.180 --> 00:00:12.960
about what does it mean to take vector A

00:00:12.960 --> 00:00:15.630
and add to that vector B.

00:00:15.630 --> 00:00:19.265
And as we'll see, we'll
get another third vector.

00:00:19.265 --> 00:00:23.490
And there's two ways that we
can think about this visually.

00:00:23.490 --> 00:00:25.520
One way is to say, all right,

00:00:25.520 --> 00:00:30.120
if we want start with vector
A and then add vector B to it,

00:00:30.120 --> 00:00:33.440
what we can do, let me
take a copy of vector B

00:00:33.440 --> 00:00:36.970
and put its tail right
at the head of vector A.

00:00:36.970 --> 00:00:38.810
Notice I have not changed the magnitude

00:00:38.810 --> 00:00:40.650
or the direction of vector B.

00:00:40.650 --> 00:00:43.850
If I did, I would actually
be changing the vector.

00:00:43.850 --> 00:00:47.270
And when I do it like that,
this defines a third vector

00:00:47.270 --> 00:00:49.680
which can be use the sum of a plus B.

00:00:49.680 --> 00:00:54.010
And the sum is going to
start at the tail of vector A

00:00:54.010 --> 00:00:57.240
and end at the head of vector B here.

00:00:57.240 --> 00:00:58.200
So let me draw that.

00:00:58.200 --> 00:01:00.680
So it would look something like that.

00:01:00.680 --> 00:01:03.360
And we can call this
right over here, vector C.

00:01:03.360 --> 00:01:08.360
So we could say A plus
B is equal to vector C.

00:01:08.450 --> 00:01:09.970
Now we could have also thought about it

00:01:09.970 --> 00:01:11.540
the other way around.

00:01:11.540 --> 00:01:15.010
We could have said,
let's start with vector B

00:01:15.010 --> 00:01:19.430
and then add vector A to that.

00:01:19.430 --> 00:01:22.220
So I'll start with the tail of vector B

00:01:22.220 --> 00:01:25.070
and then at the head of vector
B, I'm going to put the tail

00:01:25.070 --> 00:01:26.250
of vector A.

00:01:26.250 --> 00:01:29.800
So it could look something like that.

00:01:29.800 --> 00:01:33.490
And then once again, the sum
is going to have its tail

00:01:33.490 --> 00:01:34.830
at our starting point here

00:01:34.830 --> 00:01:37.440
and its head at our finishing point.

00:01:37.440 --> 00:01:39.780
Now, another way of thinking about it is

00:01:39.780 --> 00:01:42.460
we've just constructed a parallelogram

00:01:42.460 --> 00:01:45.490
with these two vectors by putting both

00:01:45.490 --> 00:01:48.450
of their tails together,
and then by taking a copy

00:01:48.450 --> 00:01:51.420
of each of them and
putting that copy's tail

00:01:51.420 --> 00:01:53.240
at the head of the other vector,

00:01:53.240 --> 00:01:55.400
you construct a parallelogram like this,

00:01:55.400 --> 00:01:57.830
and then the sum is
going to be the diagonal

00:01:57.830 --> 00:01:59.440
of the parallelogram.

00:01:59.440 --> 00:02:02.900
But hopefully you appreciate
this is the same exact idea.

00:02:02.900 --> 00:02:05.422
If you just add by
putting the head to tail

00:02:05.422 --> 00:02:08.220
of the two vectors and
you construct a triangle,

00:02:08.220 --> 00:02:10.470
the parallelogram just helps us appreciate

00:02:10.470 --> 00:02:12.140
that you can start with the yellow vector

00:02:12.140 --> 00:02:15.090
and then the blue vector
or the blue vector first

00:02:15.090 --> 00:02:16.290
and then the yellow vector.

00:02:16.290 --> 00:02:19.353
But either way, the sum is
going to be this vector C.


Cosine equation solution set in an interval

WEBVTT
Kind: captions
Language: en

00:00:00.123 --> 00:00:02.550
- [Instructor] In a previous
video, we established

00:00:02.550 --> 00:00:05.850
the entire solution set
for the following equation.

00:00:05.850 --> 00:00:09.840
And we saw that all the x's
that can satisfy this equation

00:00:09.840 --> 00:00:14.150
are a combination of these
x's and these x's here.

00:00:14.150 --> 00:00:16.350
The reason why I'm
referring to each of them

00:00:17.721 --> 00:00:21.390
as numerous xs is that for
any integer value of n,

00:00:21.390 --> 00:00:22.950
you'll get another solution.

00:00:22.950 --> 00:00:26.480
For any integer value of n,
you'll get another solution.

00:00:26.480 --> 00:00:27.860
What I wanna do in this video

00:00:27.860 --> 00:00:30.800
is to make things a
little bit more concrete.

00:00:30.800 --> 00:00:33.450
And the way that we're going
to do it is by exploring

00:00:33.450 --> 00:00:37.610
all of the x values that
satisfy this equation

00:00:37.610 --> 00:00:40.120
that sit in the closed interval

00:00:40.120 --> 00:00:44.400
from negative pi over two to zero.

00:00:44.400 --> 00:00:46.030
So I encourage you like always,

00:00:46.030 --> 00:00:48.730
pause this video and have
a go at it by yourself

00:00:48.730 --> 00:00:51.210
before we work through it together.

00:00:51.210 --> 00:00:54.240
All right, now let's work
through this together.

00:00:54.240 --> 00:00:56.330
So the first helpful thing is

00:00:56.330 --> 00:00:58.500
we have these algebraic expressions.

00:00:58.500 --> 00:01:00.720
We have things written in terms of pi.

00:01:00.720 --> 00:01:04.530
Let's approximate them
all in terms of decimals.

00:01:04.530 --> 00:01:07.950
So even pi over two, we
can approximate that.

00:01:07.950 --> 00:01:10.570
Let's see, if pi is approximately 3.14,

00:01:11.968 --> 00:01:14.363
half of that is approximately 1.57,

00:01:15.420 --> 00:01:17.030
so we could say this is approximately

00:01:17.030 --> 00:01:22.030
the closed interval from -1.57 to zero.

00:01:24.160 --> 00:01:26.790
- 1.57 isn't exactly negative pi over two,

00:01:26.790 --> 00:01:28.550
but it'll hopefully be suitable

00:01:28.550 --> 00:01:31.040
for what we're trying to do here.

00:01:31.040 --> 00:01:32.710
And now let's see if we can write

00:01:32.710 --> 00:01:34.500
the different parts of these expressions,

00:01:34.500 --> 00:01:37.600
or at least approximate them as decimals.

00:01:37.600 --> 00:01:41.060
So this could be rewritten
as x is approximately,

00:01:41.060 --> 00:01:44.210
if you were to take 1/8
times the inverse cosine

00:01:44.210 --> 00:01:46.720
of -1/6, I encourage you to verify this

00:01:46.720 --> 00:01:48.380
on your own on a calculator,

00:01:48.380 --> 00:01:51.003
you would get that that's
approximately 0.22.

00:01:53.000 --> 00:01:57.153
And then pi over four
is approximately 0.785.

00:02:00.810 --> 00:02:04.940
So this expression would
be approximately 0.22

00:02:04.940 --> 00:02:09.940
minus 0.785 times n,

00:02:11.480 --> 00:02:13.140
where n could be any integer.

00:02:13.140 --> 00:02:15.330
And then this one over here on the right,

00:02:15.330 --> 00:02:18.960
let me do that in this yellow,
x could be approximately

00:02:18.960 --> 00:02:23.960
equal to, well if this
evaluates to approximately 0.22,

00:02:25.480 --> 00:02:26.910
then this is just the negative of it,

00:02:26.910 --> 00:02:30.630
so it's going to be -0.22.

00:02:30.630 --> 00:02:33.570
And then it's plus what approximately

00:02:33.570 --> 00:02:36.993
pi over four is, so 0.785n.

00:02:39.970 --> 00:02:42.660
And now what we could do
is just try different n's

00:02:42.660 --> 00:02:45.170
and see if we're starting
above or below this interval,

00:02:45.170 --> 00:02:47.220
and then see which of
the x values actually

00:02:47.220 --> 00:02:49.230
fall in this interval.

00:02:49.230 --> 00:02:50.860
So let's just start here.

00:02:50.860 --> 00:02:53.620
If we just start at n equals zero,

00:02:53.620 --> 00:02:56.210
actually why don't I set
up a little table here,

00:02:56.210 --> 00:02:59.730
we have n here and if we
have the x value here,

00:02:59.730 --> 00:03:03.320
when n is zero, well, then
you don't see this term,

00:03:03.320 --> 00:03:06.663
and you just get approximately 0.22.

00:03:08.920 --> 00:03:11.020
Now let's compare that to the interval.

00:03:11.020 --> 00:03:13.070
The upper bound of that interval is zero.

00:03:13.070 --> 00:03:14.710
So this does not sit in the interval.

00:03:14.710 --> 00:03:18.980
So this is too high and we would want to

00:03:18.980 --> 00:03:20.610
define the x's that sit in the interval.

00:03:20.610 --> 00:03:22.470
We wanna find lower values.

00:03:22.470 --> 00:03:26.740
So it's good that here, where
you're subtracting 0.785,

00:03:26.740 --> 00:03:29.490
so I would use positive
integer values of n

00:03:29.490 --> 00:03:32.930
to decrease this 0.22 here.

00:03:32.930 --> 00:03:37.690
So when n equals one, we would
subtract 0.785 from that,

00:03:37.690 --> 00:03:39.800
and I'll round all of these
to the hundredths place,

00:03:39.800 --> 00:03:44.213
and that would get us to -0.57,

00:03:45.860 --> 00:03:47.910
and that does sit in the interval.

00:03:47.910 --> 00:03:48.840
So this looks good.

00:03:48.840 --> 00:03:50.340
So this would be a solution

00:03:50.340 --> 00:03:52.770
in that interval right over here.

00:03:52.770 --> 00:03:54.580
And let's try n equals two.

00:03:54.580 --> 00:03:58.070
So we would subtract 0.785 again,

00:03:58.070 --> 00:04:01.273
and that would get us to -1.35,

00:04:05.130 --> 00:04:09.640
not 25, 35, and that also

00:04:09.640 --> 00:04:10.740
sits in the interval.

00:04:10.740 --> 00:04:14.410
It's larger than -1.57,
so that looks good.

00:04:14.410 --> 00:04:19.030
Let's subtract 0.785
again, when n equals three,

00:04:19.030 --> 00:04:22.790
that would get us -2.14.

00:04:22.790 --> 00:04:24.440
Well, that's all of a
sudden out of the interval

00:04:24.440 --> 00:04:26.860
because that's below the lower bound here.

00:04:26.860 --> 00:04:29.690
So this is too low.

00:04:29.690 --> 00:04:33.350
So using this expression,
we've been able to find

00:04:33.350 --> 00:04:38.000
two x values that sit in the
interval that we care about.

00:04:38.000 --> 00:04:41.870
Now let's use these x
values right over here

00:04:41.870 --> 00:04:44.150
and I'll set up another table.

00:04:44.150 --> 00:04:47.750
So, let's see we have our n
and then we have our x values.

00:04:47.750 --> 00:04:50.370
So let's start with n equals
zero 'cause that's easy

00:04:50.370 --> 00:04:52.160
to compute, and then
this term would go away,

00:04:52.160 --> 00:04:56.560
and we'd have -0.22, and that's actually

00:04:56.560 --> 00:04:58.370
in this interval here, it's below zero,

00:04:58.370 --> 00:05:02.660
it's larger than -1.57,
so that one checks out.

00:05:02.660 --> 00:05:05.880
But now to really explore, we
have to go in both directions.

00:05:05.880 --> 00:05:08.610
We have to increase it or decrease it.

00:05:08.610 --> 00:05:10.110
So if we wanted to increase it,

00:05:10.110 --> 00:05:12.760
we could have a situation
where n equals one.

00:05:12.760 --> 00:05:16.740
So if n equals one, we're
gonna add 0.785 to this.

00:05:16.740 --> 00:05:18.290
Now you immediately know
that that's going to be

00:05:18.290 --> 00:05:22.130
a positive value, if you
computed it, it'd be 0.57,

00:05:22.130 --> 00:05:26.600
which is larger than
zero, so this is too high.

00:05:26.600 --> 00:05:30.180
So now we could try going lower than -0.22

00:05:30.180 --> 00:05:32.500
by having negative values of n.

00:05:32.500 --> 00:05:37.060
So if n is equal to -1, that
means we're subtracting 0.785

00:05:37.060 --> 00:05:41.653
from this right over here
which would get us to -1.01.

00:05:44.870 --> 00:05:47.780
Well, that one works out,
so that's in our interval.

00:05:47.780 --> 00:05:50.570
And now let's subtract 0.785 again.

00:05:50.570 --> 00:05:53.020
So I'll have n equals -2.

00:05:53.020 --> 00:05:57.410
And so if I subtract 0.785 again,

00:05:57.410 --> 00:06:01.413
I could round that to -1.79,

00:06:02.270 --> 00:06:04.640
which is lower than -1.57,

00:06:04.640 --> 00:06:08.200
so it's out of our
interval, so it's too low.

00:06:08.200 --> 00:06:11.050
So all of the x values
that are in our interval

00:06:11.050 --> 00:06:15.020
that satisfy this equation
are these two right over here.

00:06:15.020 --> 00:06:19.410
And this one and this one,

00:06:19.410 --> 00:06:20.923
and we are done.