Question on Vectors

Question

Given $a = [-5, 8, 1]$ and $b = [2, -7, -3]$, find a vector $c$ such that $a \cdot (b × c) = 0$

I don't know how to get it, I've been looking for examples, but I don't know..

Answer 1

I like anything in the span of $\mathbf{a}$ and $\mathbf{b}$ so anything of the form:

$$\mathbb{c}=k\, \mathbf{a}+\lambda\, \mathbf{b},$$ for scalars $k$ and $\lambda$.

If you draw a picture, you will see that in this case $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are all in the same plane and so $\mathbf{b}\times\mathbf{c}$ will be perpendicular to the plane and particular to $\mathbf{a}$.

Also, algebraically,

$$\mathbf{b}\times\mathbf{c}=\mathbf{b}\times (k\,\mathbf{a}+\lambda \,\mathbf{b})=k\,\mathbf{b}\times \mathbf{a}+\lambda\,\mathbf{b}\times\mathbf{b}=k\,\mathbf{b}\times \mathbf{a},$$

and now

$$\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\mathbf{a}\cdot (k\,\mathbf{b}\times \mathbf{a})=k\,(\mathbf{a}\cdot(\mathbf{b}\times\mathbf{a}))=0.$$

Answer 2

Let c=(0,0,0). Not probably what you meant...