vector product and scalar product equations

Question

Find $x$ given that $x \times a = b$ and $x \cdot c = d$ where $a \cdot c$ is not $0$.

Where $\times$ is the cross product and $\cdot$ is the dot product.

I have tried to manipulate this in more ways but everything led to a dead end, any starting point suggestions?

Answer 1

Start with the vector triple product identity

$$ A \times (B \times C) = (A\cdot C ) B - (A\cdot B) C $$

This means that

$$ c \times (x \times a) + (c \cdot x) a = (c\cdot a) x $$

solving we get

$$ x = \frac{1}{a\cdot c} \left( c \times b + d a\right) $$

Answer 2

I suppose $x=(x,y,z)$ and $a=(a_1,a_2,a_3),b=(b_1,b_2,b_3),c=(c_1,c_2,c_3) \in \mathbb{R}^3$ (known vectors), and $d\in \mathbb{R}$ , so we have the system of equations: $$ \begin{cases} a_3y-a_2z=b_1\\ a_3y-a_1z=b_2\\ a_2x-a_1y=b_3\\ c_1x+c_2y+c_3z=d \end{cases} $$

can you solve it?