Mathematics

Question

Find the cross product of [tex]- \frac{3}{4}v[/tex] and [tex]- \frac{1}{2} w[/tex] if [tex]v=[-2,12,-3][/tex] and [tex]w=[-7,4,-6][/tex]

1 Answer

  • For any scalars [tex]c_1,c_2[/tex], we have

    [tex]c_1\mathbf v\times c_2\mathbf w=c_1c_2\mathbf v\times\mathbf w[/tex]

    So

    [tex]\left(-\dfrac34\mathbf v\right)\times\left(-\dfrac12\mathbf w\right)=\dfrac38\mathbf v\times\mathbf w[/tex]

    We have

    [tex]\mathbf v\times\mathbf w=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\-2&12&-3\\-7&4&-6\end{vmatrix}[/tex]
    [tex]=\begin{vmatrix}12&-3\\4&-6\end{vmatrix}\mathbf i-\begin{vmatrix}-2&-3\\-7&-6\end{vmatrix}\mathbf j+\begin{vmatrix}-2&12\\-7&4\end{vmatrix}\mathbf k[/tex]
    [tex]=-60\,\mathbf i-(-9)\,\mathbf j+76\,\mathbf k[/tex]
    [tex]=\begin{bmatrix}-60\\9\\76\end{bmatrix}[/tex]

    which makes

    [tex]\left(-\dfrac34\mathbf v\right)\times\left(-\dfrac12\mathbf w\right)=\dfrac38\begin{bmatrix}-60\\9\\76\end{bmatrix}=\begin{bmatrix}-\frac{45}2\\\\\frac{27}8\\\\\frac{57}2\end{bmatrix}[/tex]
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