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}* \vec{B}`

To proof =

`\vec{A}* \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