$$\ddot{\gamma} (t) \times \dot{\gamma} (t)=(-a\cos t, -a\sin t, 0)\times (-a\sin t, a\cos t, b)$$ The writer will get the following result $$(-ab\sin t, ab\cos t, -a^2)$$
but I don't know how to compute the product. Please teach me this step by step clearly and explanatorily. Thank you for help.
$$\begin{align} & (-a \cos t, -a \sin t, 0) \times (-a \sin t, a \cos t, b)\\ \\ = & \begin{vmatrix} \hat{x} & \hat{y} & \hat{z}\\ -a \cos t & -a \sin t & 0 \\ -a \sin t & a \cos t & b \end{vmatrix}\\ \\ = & \hat{x}( b(-a \sin t)-0)-\hat{y} (-ab\cos t-0) +\hat{z}(-a^2 \cos^{2}t-a^2 \sin^{2}t)\\ = & \hat{x}(-ab\sin t)+\hat{y}(ab\cos t)+\hat{z}(-a^2(\cos^{2}t+\sin^{2}t))\\ = & (-ab\sin t,ab\cos t,-a^2) \end{align}$$ since $\sin^{2}t+\cos^{2}t=1$.