Chapter 2.4, Problem 227E

Let $u=\langle u_1,u_2,0\rangle$ and $v=\langle v_1,v_2,0\rangle ,$ be two-dimensional vector. The cross product of vectors $u$ and $v$ is net defined. However, if the vectors are regarded as the three-dimensional vectors $u=\langle u_1,u_2,0\rangle$ and $v=\langle v_1,v_2,0\rangle ,$ respectively, then, in this case, we can define the cross product of $u$ and $v$ . In particular, in determinant notation, the cross product of $u$ and $v$ is given by

$u\times v=\left|\begin{array}{ccc}i& j& k\\ u_1& u_2& 0\\ v_1& v_2& 0\end{array}\right|.$

Use this result to compute $\left(i\cos \theta +j\sin \theta \right)\times \left(i\sin \theta -j\cos \theta \right)$ , where $\theta$ is a real number.