Answer
(a) r x s = i + 4j + k
(b) The magnitude of r x s is √18
Explanation
Given vectors:
r = 3i - 2j + 5k
s = -2i - j + 2k
Step-by-step solution:
(a) Find r x s
[tex]\begin{gathered} r\times s=\begin{bmatrix}{i} & {j} & {k} \\ {3} & {-2} & {5} \\ {2} & {-1} & {2}\end{bmatrix}=i\begin{bmatrix}{-2} & {5} \\ {-1} & {2}\end{bmatrix}-j\begin{bmatrix}{3} & {5} \\ {2} & {2}\end{bmatrix}+k\begin{bmatrix}{3} & {-2} \\ {2} & -{1}\end{bmatrix} \\ =i((-2\times2)-(5\times-1))-j((3\times2)-(5\times2))+k((3\times-1)-(-2\times2)) \\ \\ =i(-4+5)-j(6-10)+k(-3+4) \\ \\ =i+4j+k \end{gathered}[/tex]Therefore, r x s = i + 4j + k
(b) Find the magnitude of r x s
[tex]\lvert{r\times s}\rvert=\lvert{i+4j+k}\rvert=\sqrt{1^2+4^2+1^2}=\sqrt{1+16+1}=\sqrt{18}[/tex]Hence, the magnitude of r x s is √18
