Question

Evaluate and simplify the indicated quantity. ((sin t)i + (cos t)j + tk) × ((cos t)i + (sin t)j)

Answer 1

let us consider

$\vec{A} = sint\hat{i}+cost\dot{j}+ t\hat{k}\ ,\vec{B} =cost\hat{i} + sint\hat{j}$

now we have to evaluate

$\vec{A}\times \vec{B}$

To proof =

$\vec{A}\times \vec{B}=\begin{vmatrix}i &j &k \\ sint&cost &t \\ cost&sint &0 \end{vmatrix}$

now solving the determinant form

we get

=$\hat{i}\left ( -sint \right )-\hat{j}\left ( -tcost \right )+\hat{k}\left ( sin^{2}t - cos^{2}t\right )$

by using the formula

$sin^{2}t + cos^{2}t = 1$

$cos^{2}t = sin^{2} t - 1$

put this value in the above equation

we get

=$\hat{i}\left ( -sint \right )+\hat{j}\left ( tcost \right )+\hat{k}\left ( 2sin^{2}t - 1\right )$

Hence proved