Answer to Question #209431 in Analytic Geometry for Tshego

Question

Let u=<-2,1,-1, V =< - 3,2 - 1> and w=<1,3,5>. Compute :

a.) u x w,

b.) u x (w x v) and (u x w) x v.

Answer 1

Let $u=<-2,1,-1>, v=<-3,2,-1>, w=<1,3,5>$.

Compute:

a) $u\times w= \begin{vmatrix} i &j &k \\ -2&1&-1\\ 1&3&5 \end{vmatrix}=i (1·5 - (-1)·3) - j ((-2)·5 - (-1)·1) + k ((-2)·3 - 1·1) = i (5 + 3) - j (-10 + 1) + k (-6 - 1) = <8; 9; -7>$

b)

$u\times (w\times v)=\begin{vmatrix} i&j&k \\ 1&3&5\\ -3&2&-1 \end{vmatrix}=i (3·(-1) - 5·2) - j (1·(-1) - 5·(-3)) + k (1·2 - 3·(-3)) = i (-3 - 10) - j (-1 + 15) + k (2 + 9) =$

$=\begin{vmatrix} i&j&k \\ -2&1&-1\\ -13&-14&11 \end{vmatrix}= i (1·11 - (-1)·(-14)) - j ((-2)·11 - (-1)·(-13)) + k ((-2)·(-14) - 1·(-13)) = i (11 - 14) - j (-22 - 13) + k (28 + 13)= <-3; 35; 41>$

$(u×w)×v=\begin{vmatrix} i&j&k \\ -2&1&-1\\ 1&3&5 \end{vmatrix}=i (1·5 - (-1)·3) - j ((-2)·5 - (-1)·1) + k ((-2)·3 - 1·1) = i (5 + 3) - j (-10 + 1) + k (-6 - 1) = \begin{vmatrix} i&j&k\\ 8&9&-7\\ -3&2&-1 \end{vmatrix}=i (9·(-1) - (-7)·2) - j (8·(-1) - (-7)·(-3)) + k (8·2 - 9·(-3)) = i (-9 + 14) - j (-8 - 21) + k (16 + 27) = <5; 29; 43>.$