Question #210884

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.


1
Expert's answer
2021-06-29T12:30:19-0400

6.1) u×w=ijk211135u×w=\begin{vmatrix} i & j &k \\ -2 & 1 & -1\\ 1 & 3 & 5 \end{vmatrix} =i(5+3)j(10+1)+k(61)=8i+9j7k=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+35)<3,2,1>=<9,27,45><12,8,4>=<3,35,41>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> (u×w)×v=<8,9,7>×<3,2,1>=ijk897321\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(821)+k(16+27)=5i+29j+43k=i(-9+14)-j(-8-21)+k(16+27) =5i+29j+43k


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