Question

Evaluate and simplify the indicated quantities in Exercises 35–41.
$\left(\right(\sin t)i+(\cos t)j+tk)\times \left(\right(\cos t)i+(\sin t\left)j\right)$

Short answer

The simplification of $\left(\right(\sin t)i+(\cos t)j+tk)\times \left(\right(\cos t)i+(\sin t\left)j\right)$is $(-t\sin t,t\cos t,{\sin }^{2}t-{\cos }^{2}t)$.

Step-by-step solution

Step 1. Given Information.

The quantities:

$\left(\right(\sin t)i+(\cos t)j+tk)\times \left(\right(\cos t)i+(\sin t\left)j\right)$

Step 2. Do the cross product.

By Cross product,

$$\begin{gathered} \left(\right(\sin t)i+(\cos t)j+tk)\times \left(\right(\cos t)i+(\sin t\left)j\right)=\left(\right(\cos t\left)\right(0)-(\sin t)t,(\cos t)t-(\sin t)0,(\sin t\left)\right(\sin t)-(\cos t\left)\right(\cos t\left)\right) \\ =(-t\sin t,t\cos t,{\sin }^{2}t-{\cos }^{2}t) \end{gathered}$$