Chapter 11, Problem 26P

(II) (a) Show that the cross product of two vectors, $\stackrel{\to }{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}$, and $\stackrel{\to }{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}$is

$\stackrel{\to }{A}\times \stackrel{\to }{B}=\left(A_y{B}_z-A_z{B}_y\right)\hat{i}+\left(A_z{B}_x-A_x{B}_z\right)\hat{j}+\left(A_x{B}_y-A_y{B}_x\right)\hat{k}$.

(b) Then show that cross product can be written

$\stackrel{\to }{A}\times \stackrel{\to }{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ A_x& A_y& A_z\\ B_x& B_y& B_z\end{array}\right|$,

where we use the rules for evaluating a determinant. (Note, however, that this is not really a determinant, but a memory aid.)