Answer to Question #210884 in Analytic Geometry for Ree

Question

Question 6

Let u =< −2,1,−1,v =< −3,2,−1 >and w =< 1,3,5 >. Compute:

(6.1) u ×w,

(6.2)u ×(w ×v)and(u × w)×v.

Answer 1

6.1) $u×w=\begin{vmatrix} i & j &k \\ -2 & 1 & -1\\ 1 & 3 & 5 \end{vmatrix}$ $=i(5+3)-j(-10+1)+k(-6-1) =8i+9j-7k$ $6.2) u×(w×v)=(u.v)w-(u.w)v ={<-2,1,-1>.<-3,2,-1>}<1,3,5>-{<-2,1,-1>.<1,3,5>}<-3,2,-1> =(6+2+1)<1,3,5>-(-2+3-5)<-3,2,-1> =<9,27,45>-<12,-8,4>=<-3,35,41>$ $\to (u×w)×v=<8,9,-7>×<-3,2,-1>=\begin{vmatrix} i & j &k \\ 8 & 9 & -7\\ -3 & 2 & -1 \end{vmatrix}$$=i(-9+14)-j(-8-21)+k(16+27) =5i+29j+43k$