외적

Let $\text{u}=\langle {u}_{1},{u}_{2}\rangle$ and $\text{v}=\langle {v}_{1},{v}_{2}\rangle$ be two-dimensional vectors. The cross product of vectors $\text{u}$ and $\text{v}$ is not defined. However, if the vectors are regarded as the three-dimensional vectors $\tilde{\text{u}}=\langle {u}_{1},{u}_{2},0\rangle$ and $\tilde{\text{v}}=\langle {v}_{1},{v}_{2},0\rangle ,$ respectively, then, in this case, we can define the cross product of $\tilde{\text{u}}$ and $\tilde{\text{v}}.$ In particular, in determinant notation, the cross product of $\tilde{\text{u}}$ and $\tilde{\text{v}}$ is given by

$$\tilde{\text{u}}\,\times \,\tilde{\text{v}}=|\begin{array}{ccc}\text{i} & \text{j} & \text{k} \\ {u}_{1} & {u}_{2} & 0 \\ {v}_{1} & {v}_{2} & 0\end{array}|.$$

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