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uysha [10]
3 years ago
8

Find the cross product of

alt="- \frac{3}{4}v" align="absmiddle" class="latex-formula"> and - \frac{1}{2} w if v=[-2,12,-3] and w=[-7,4,-6]
Mathematics
1 answer:
dsp733 years ago
4 0
For any scalars c_1,c_2, we have

c_1\mathbf v\times c_2\mathbf w=c_1c_2\mathbf v\times\mathbf w

So

\left(-\dfrac34\mathbf v\right)\times\left(-\dfrac12\mathbf w\right)=\dfrac38\mathbf v\times\mathbf w

We have

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

which makes

\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}
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