$a^Tx=b$ denotes a hyperplane. Is it always true that $a//x$

Question

$a^Tx=b$ denotes a hyperplane. Is it always true that $a//x$? $x \in R^n$ $a\in R^n$ anb $b \in R$.From this data can we infer that $\vec a$ is parallel to $\vec x$ and if yes why?

Answer 1

There is no reasons for ${\bf{a}}$ and ${\bf{x}}$ to be parallel for all ${\bf{x}}$ satisfying ${{\bf{a}}^T}{\bf{x}} = b$. However, suppose we want to find one ${\bf{x}}$ which satisfies ${{\bf{a}}^T}{\bf{x}} = b$ and is parallel to ${\bf{a}}$ which means ${\bf{x}} = \alpha {\bf{a}}$. So we have

$$\eqalign{ & {\bf{x}} = \alpha {\bf{a}} \cr & {{\bf{a}}^T}{\bf{x}} = \alpha {{\bf{a}}^T}{\bf{a}} = \alpha {\left\| {\bf{a}} \right\|^2} \cr & \alpha = {{{{\bf{a}}^T}{\bf{x}}} \over {{{\left\| {\bf{a}} \right\|}^2}}} = {b \over {{{\left\| {\bf{a}} \right\|}^2}}} \cr} \tag{1}$$

So if we choose $\alpha $ in the way above, ${\bf{x}}$ will satisfy the equation of the hyperplane and is also parallel to ${\bf{a}}$. Finally, the special ${\bf{x}}$ having this property is given by

$${\bf{x}} = {b \over {{{\left\| {\bf{a}} \right\|}^2}}}{\bf{a}}\tag{2}$$