Prove that vectors u, v and w are coplanar if and only if vectors u, v and w are linearly dependent.

Answer 1

normal to plane of u and v =
i j k
ux uy uz
vx vy vz

UV = i(uy vz - vy uz)+j(uz vx - ux vz)+k (ux vy - uy vx)

normal to plane of u and w
UW =i(uy wz -wy uz) +j(uz wx - ux wz) +k (ux wy -uy wx)

now if those vectors are in the same direction then UW = k UV