Vectors — Maths STD 12 Science — Question

Let $\bar{a}, \bar{b}, \& \bar{c}$ be position vectors of $A, B \& C$
$\overline{ AB }=\overline{ b }-\overline{ a }$
$=(-2 \hat{i}+\hat{j})-(-7 \hat{i}+4 \hat{j}-2 \hat{k})$
$=5 \hat{ i }-3 \hat{ j }+2 \hat{ k }$
$\overline{ AC }=\overline{ c }-\overline{ a }$
$=(3 \hat{ i }-2 \hat{ j }+2 \hat{ k })-(-7 \hat{ i }+4 \hat{ j }-2 \hat{ k })$
$=10 \hat{ i }-6 \hat{ j }+4 \hat{ k }$
$=2[5 \hat{ i }-3 \hat{ j }+2 \hat{ k }]$
$\Rightarrow \overline{ AC }=2 \overline{( AB )}$
$\Rightarrow \overline{ AC }$ is a scalar multiple of $\overline{ AB }$
$\Rightarrow \overline{ AC } \& \overline{ AB }$
∵ A is common
$\Rightarrow A B \& C$ are collinear.