# Scalar triple product coplanarity

## Homework Statement

Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0.

## The Attempt at a Solution

if u · (v × w) = 0, then u is orthogonal to vxw, and
vxw is orthogonal to v and w.

therefore, u must lie in the same plane determined by v and w (i.e. they are coplanar)

Is this correct? Also, how would i describe this proof mathematically?

Thanks!

## Answers and Replies

u · (v × w) = 0

use geometrical proof

translate the equation as volume of parallopiped
the volume is zero only when the height is zero
or all vectors lie in the same plane

HallsofIvy
Science Advisor
Homework Helper

## Homework Statement

Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0.

## The Attempt at a Solution

if u · (v × w) = 0, then u is orthogonal to vxw, and
vxw is orthogonal to v and w.

therefore, u must lie in the same plane determined by v and w (i.e. they are coplanar)

Is this correct? Also, how would i describe this proof mathematically?

Thanks!
Yes, that's correct and, once you have stated exactly HOW you know that two vectors, perpendicular to the same non-zero vector, are co-planar, it IS "mathematical".

payumooli's suggestion is another way to do it but since you have already done it your own way, I suggest you stay with it.