Find the cross product of - (3)/(4)v and - (1)/(2) w if v=[-2,12,-3] and w=[-7,4,-6]

Question

Find the cross product of
`- (3)/(4)v` and
`- (1)/(2) w` if
`v=[-2,12,-3]` and
`w=[-7,4,-6]`

Answer 1

For any scalars
`c_1,c_2`, we have


`c_1\mathbf v* c_2\mathbf w=c_1c_2\mathbf v*\mathbf w`

So


`\left(-\frac34\mathbf v\right)*\left(-\frac12\mathbf w\right)=\frac38\mathbf v*\mathbf w`

We have


`\mathbf v*\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(-\frac34\mathbf v\right)*\left(-\frac12\mathbf w\right)=\frac38\begin{bmatrix}-60\\9\\76\end{bmatrix}=\begin{bmatrix}-\frac{45}2\\\\\frac{27}8\\\\\frac{57}2\end{bmatrix}`