Vector equation

Let $\mathbf{a, b, c} \in \mathbb{R}^3$ be three vectors. Solve the equation:

$$\mathbf{x+(x\cdot a)b = c}$$

Solution

Successively we have:

$$\begin{align*}
\mathbf{x+ (x\cdot a)b=c} &\Rightarrow   \mathbf{x \cdot a+ (x\cdot a)\cdot (a\cdot b)=a\cdot c}  \\
 &\Rightarrow \mathbf{x\cdot a= \frac{a\cdot c}{1+a\cdot b}} \\
 &\Rightarrow  \mathbf{x=c - \frac{a\cdot c}{1+a\cdot b}b}
\end{align*}$$

The last solution verifies the initial if an only if $1+ \mathbf{a\cdot b} \neq 0$. We are investigating further.

Investigation 

If $\mathbf{1+a\cdot b}=0$ then $\mathbf{x}=m\mathbf{b}+n \mathbf{c}$ so the initial equation is transfmored into:

$$m\mathbf{b}+n \mathbf{c}+ \left ( -m +n \cdot \mathbf{a\cdot c} \right )\cdot \mathbf{b}=\mathbf{c}$$

or in a more simplified form: $\displaystyle n \mathbf{c}+ n \mathbf{(a\cdot c)\cdot b}=\mathbf{c}$.

• If $\mathbf{a \cdot c}=0$ then $n=1$ hence $\mathbf{x}=m \mathbf{b} +\mathbf{c} , \;\; m \in \mathbb{R}$.
• If $\mathbf{a \cdot c}\neq 0$ and $\mathbf{b, c}$ not collinear then the initial equation is obviously impossible .
• If $\mathbf{a\cdot c} \neq 0$ and $\mathbf{b, c}$ are collinear then the equation is transformed to: $$ k \mathbf{b}+ \left ( k \mathbf{b \cdot a} \right )\cdot b = \ell \mathbf{b}\Rightarrow \ell \mathbf{b}=0 \Rightarrow \mathbf{b}=0 \; {\rm or} \; \mathbf{c}=0$$