Solving vector equations. (Vectors are not with components)

Question

How to solve this and what is the answer?

Answer 1

HINT

Observe that

$$\alpha \vec x + \vec a \times \vec x = \vec b \implies \vec a \cdot(\alpha \vec x + \vec a \times \vec x) = \vec a \cdot\vec b \implies \alpha\,\vec a\cdot \vec x = \vec a \cdot \vec b$$

We can use this to verify which solution holds.

Answer 2

First, I'm assuming that these vectors are three-dimensional. Write $x = (x_1,x_2,x_3)$ and similarly for $a$ and $b$. Calculate $\alpha x$ and $a\times x$, and add them together to get a vector with three components that we can set equal to $b$'s three components. The equations will each be in the form of $c_1 x_1 + c_2 x_2 + c_3 x_3 = c_4$ for some constants $c_i$, which will give three linear equations with three unknowns. We can solve this with whatever techniques you have for that, which will give you the components of $x$ in terms of the components of $a$ and $b$. Then you can try to simplify the three components into a single vector expression, which will be your answer.