How to prove three vectors are in same plane?

Question

I know that three vectors lie on the same plane if and only if u.(v×w)=0. (VxW) gives us a vector which is perpendicular to the plane.But if u vector is on another plane which plane is parallel to (VxW) plane still we get u.(V×W)=0.Then how can i be sure about that three vectors are in same plane?

Answer 1

Hint: Assume that $u$ and $v$ are not proportional (or else there's nothing to do), and write $w$ as a linear combination of $u$, $v$ and $u\times v$. Check the coefficient in front of $u\times v$ is precisely $u\cdot (v\times w)$.

Answer 2

If you're dealing with a three-dimensional Euclidean vector space -- the kind of vector space where it makes sense to say $V\times W$ is perpendicular to $V$ and $W$ -- then a vector is just a direction and distance (or length) in three dimensions.

If a vector $u$ is parallel to a vector $u'$, in the same direction as $u'$, and is the same length as $u'$, then $u = u'.$

So what is this vector $u$ that is parallel to the plane of $v$ and $w$ but is not in the plane of $v$ and $w$?

You may think that the vector $u$ can be outside the $v, w$ plane if you put its starting point outside the $v, w$ plane. But "starting point" is not a distinguishing property of vectors in this vector space. Only direction and length can make two vectors different.