Question

Three points $(2, -1,3)$, $(3, -5,1)$ and $(-1, 11, 9)$ are

• A. Non-collinear
• B. Non-coplanar
• C. Collinear
• D. None of these

Answer 1

Answer: (C)

Let $A$, $B$ and $C$ be three points whose coordinates are $(2, -1,3)$, $(3, -5,1)$ and $(-1,11,9)$ respectively, then
$\overrightarrow{OA} = 2\hat{i} - \hat{j} +3\hat{k}$,
$ \overrightarrow{OB} = 3\hat{i} - 5\hat{j} +\hat{k}$
and $\overrightarrow{OC} = -\hat{i} -11 \hat{j} +9\hat{k}$
$\therefore \overrightarrow{AB}= \overrightarrow{OB} - \overrightarrow{OA} = \left( 3\hat{i} - 5\hat{j} +\hat{k}\right) - \left( 2\hat{i} - \hat{j} +3\hat{k}\right)$
$= \hat{i} - 4\hat{j} - 2\hat{k}$
$ \overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA} = \left( -\hat{i} -11 \hat{j} +9\hat{k}\right) - \left( 2\hat{i} - \hat{j} +3\hat{k}\right)$
$= - 3\hat{i} +12 \hat{j} +6\hat{k}$
$\Rightarrow \overrightarrow{AC} =-3 \overrightarrow{AB}$
Thus, the vector $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel having the same initial point $A$.
Hence, the points $A$, $B$, $C$ are collinear.