# Cross product of $\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle$ and $\left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle$

The calculator will find the cross product of two vectors $\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle$ and $\left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle$, with steps shown.
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Calculate $\left\langle - \frac{\sin{\left(t \right)}}{2}, \frac{\sqrt{3}}{2}, \frac{\cos{\left(t \right)}}{2}\right\rangle\times \left\langle - \cos{\left(t \right)}, 0, - \sin{\left(t \right)}\right\rangle.$

### Solution

To find the cross product, we form a formal determinant the first row of which consists of unit vectors, the second row is our first vector, and the third row is our second vector: $\left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\- \frac{\sin{\left(t \right)}}{2} & \frac{\sqrt{3}}{2} & \frac{\cos{\left(t \right)}}{2}\\- \cos{\left(t \right)} & 0 & - \sin{\left(t \right)}\end{array}\right|$.

Now, just expand along the first row (for steps in finding a determinant, see determinant calculator):

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

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

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