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

Question

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

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    2022-01-04T19:17:44+00:00

    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

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