# Geometry problem involving dot/cross product

1. Sep 28, 2009

### hanelliot

1. The problem statement, all variables and given/known data
Let r be a line and pi be a plane with equations
r: P + tv
pi: Q + hu + kw (v, u, w are vectors)
Assume v · (u x w) = 0. Show that either r ∩ pi = zero vector or r belongs to pi.

2. Relevant equations
n/a

3. The attempt at a solution
I get the basic idea behind it but I'm not sure if my "solution" is formally good enough. I know that cross product gives you a normal to a plane, which is u x w here. I also know that the dot product = 0 means that they are perpendicular to each other.. so v and (u x w) are perpendicular to each other. If you draw it out, you can clearly see that r must be either in pi or out of pi. Is this good enough to warrant a good mark? Thanks!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 28, 2009

### lanedance

if r is parallel to pi, but not "in" pi their intersection will not be the zero vector, it will be the empty set

try the separate cases when P is "in" pi, and when P is not "in" pi and try and see if you can show whether any arbitrary point on r is in, or not in P, knowing that v = a.(uXw) where a is some non-zero constant

3. Sep 29, 2009

### hanelliot

you are right, my mistake

I can show that with intuition and a sketch of plane/line but not sure how I should go about proving it formally.. maybe this Q is that simple and I'm overreacting

4. Sep 29, 2009

### lanedance

as you know
v.(u x w) = 0

then
v = au + bw
for constants a & b - why?

equation of you line is P + vt

now, i haven't tried, but using your equation of a line & the equation of a plane, considering the following cases, should show what you want:

Case 1 - P is a point in pi. Now show any other point on the line, using the line equation, satisfies the equation defining pi.

Case 2 - P is not a point in pi. Now show any other point on the line, using the line equation, does not satisfy the equation defining pi.