Producto cruzado de $$$\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle$$$ y $$$\left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle$$$

Calcular $$$\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle\times \left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle.$$$

Para encontrar el producto vectorial, formamos un determinante formal cuya primera fila consta de vectores unitarios, la segunda fila es nuestro primer vector y la tercera fila es nuestro segundo vector: $$$\left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\- \frac{\sin{\left(t \right)}}{2} & \frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\- \cos{\left(t \right)} & 0 & - \sin{\left(t \right)}\end{array}\right|$$$.

Ahora, simplemente expanda a lo largo de la primera fila (para conocer los pasos para encontrar un determinante, consulte calculadora de determinantes):

$$$\left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\- \frac{\sin{\left(t \right)}}{2} & \frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\- \cos{\left(t \right)} & 0 & - \sin{\left(t \right)}\end{array}\right| = \left|\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\0 & - \sin{\left(t \right)}\end{array}\right| \mathbf{\vec{i}} - \left|\begin{array}{cc}- \frac{\sin{\left(t \right)}}{2} & \frac{\cos{\left(t \right)}}{2}\\- \cos{\left(t \right)} & - \sin{\left(t \right)}\end{array}\right| \mathbf{\vec{j}} + \left|\begin{array}{cc}- \frac{\sin{\left(t \right)}}{2} & \frac{\sqrt{3}}{2}\\- \cos{\left(t \right)} & 0\end{array}\right| \mathbf{\vec{k}} = \left(\left(\frac{\sqrt{3}}{2}\right)\cdot \left(- \sin{\left(t \right)}\right) - \left(\frac{\cos{\left(t \right)}}{2}\right)\cdot \left(0\right)\right) \mathbf{\vec{i}} - \left(\left(- \frac{\sin{\left(t \right)}}{2}\right)\cdot \left(- \sin{\left(t \right)}\right) - \left(\frac{\cos{\left(t \right)}}{2}\right)\cdot \left(- \cos{\left(t \right)}\right)\right) \mathbf{\vec{j}} + \left(\left(- \frac{\sin{\left(t \right)}}{2}\right)\cdot \left(0\right) - \left(\frac{\sqrt{3}}{2}\right)\cdot \left(- \cos{\left(t \right)}\right)\right) \mathbf{\vec{k}} = - \frac{\sqrt{3} \sin{\left(t \right)} \mathbf{\vec{i}}}{2} - \frac{\mathbf{\vec{j}}}{2} + \frac{\sqrt{3} \cos{\left(t \right)} \mathbf{\vec{k}}}{2}$$$

Por lo tanto, $$$\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle\times \left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle = \left\langle - \frac{\sqrt{3} \sin{\left(t \right)}}{2}, - \frac{1}{2}, \frac{\sqrt{3} \cos{\left(t \right)}}{2}\right\rangle.$$$

$$$\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle\times \left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle = \left\langle - \frac{\sqrt{3} \sin{\left(t \right)}}{2}, - \frac{1}{2}, \frac{\sqrt{3} \cos{\left(t \right)}}{2}\right\rangle\approx \left\langle - 0.866025403784439 \sin{\left(t \right)}, -0.5, 0.866025403784439 \cos{\left(t \right)}\right\rangle$$$A