Answer
Proof given below.
Work Step by Step
If the line has slope defined, then $a\neq 0$ and $y=\displaystyle \frac{b}{a}x-c$, the slope of the line is $\displaystyle \frac{b}{a}.$
The slope of ${\bf v}$=$\langle a,\ \ b \rangle$ is also $\displaystyle \frac{b}{a}$, when $a\neq 0$.
So, for $a\neq 0,\ {\bf v}$ is parallel to the line.
If $a=0$, the vector ${\bf v}$ has undefined slope (parallel to ${\bf j}).$
If $a=0$, the line $\quad bx=c\quad $is vertical, parallel to the y-axis.
${\bf v}$ is parallel to the line in this case as well..
