Answer to Question #228929 in Analytic Geometry for Afnan

Question

The volume V of the parallelopiped that has u, v, w as adjacent edges is

given by: V = |u.(v × w)|. If u.(v × w) = 0, then u, v, w lie in the same

plane. Thus, find the volume of the parallelopiped formed by the followings:

u =< 1, 2, −1 >, v =< 2, −1, 2 >, w =< 2, −3, 4 >.

Answer 1

$v\times{w}=\begin{vmatrix} i & j & k \\ 2 & -1 & 2\\ 2 & -3 & 4 \end{vmatrix}=i(4(-1)-2(-3))-j(4(2)-2(2))+k(2(-3)-2(-1))=2i-4j-4k\\ u\cdot(v\times{w})=(i+2j-k)(2i-4j-4k)=2-8+4=-2\\ V=|u\cdot(v\times{w})|=|-2|=2$