# Scalar triple product coplanarity

Neen87

## 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

payumooli
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

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.