Question

Consider two vectors A and B A=14i and B= -4i+8j

* = dot product x=cross product

Find {[A*B]i}*A and {[A+B]xB}xk

Answer 1

Assuming
`\mathbf a,\mathbf b\in\mathbb R^3`, you have


`\mathbf a\cdot\mathbf b=(14)(-4)+(0)(8)=-56`

so


`((\mathbf a\cdot\mathbf b)\,\mathbf i)\cdot\mathbf a=(-56\,\mathbf i)\cdot(14\,\mathbf i)=(-56)(14)=-784`

Next,


`\mathbf a+\mathbf b=(14\,\mathbf i)+(-4\,\mathbf i+8\,\mathbf j)=10\,\mathbf i+8\,\mathbf j`

Then


`(\mathbf a+\mathbf b)*\mathbf b=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\10&8&0\\-4&8&0\end{vmatrix}=112\,\mathbf k`


`((\mathbf a+\mathbf b)*\mathbf b)*\mathbf k=\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\0&0&112\\0&0&1\end{vmatrix}=\mathbf 0`