Geometry problem involving dot/cross product

[hanelliot]

Homework Statement

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.

Homework Equations

n/a

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!

 

[lanedance]

Homework Statement

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.

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

[

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!

Homework Statement

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

 

[hanelliot]

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

you are right, my mistake

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

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

 

[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.