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  4. The points with position vectors 60 hati +3 hatj, 40 hati - 8 hatj and a hati -52 hatj are collinear if

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Q. The points with position vectors $ 60\hat{i} +3 \hat{j}$, $40\hat{i} - 8\hat{j}$ and $ a\hat{i} -52 \hat{j} $ are collinear if

Vector Algebra

Solution:

Let $\overrightarrow{OA} = 60\hat{i} +3 \hat{j}$,
$ \overrightarrow{OB} = 40\hat{i} - 8\hat{j} $
and $\overrightarrow{OC} = a\hat{i} -52 \hat{j} $
$ \overrightarrow{AB}= \overrightarrow{OB} - \overrightarrow{OA} = 20\hat{i} - 11\hat{j} $
$ \overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA} = \left(a-60\right)\hat{i} -55 \hat{j} $
Now, $\overrightarrow{AB} \propto \overrightarrow{AC}$ as $A$, $B$, $C$ are collinear.
i.e. $\left( - 20\hat{i} - 11\hat{j} \right) = \lambda\left\{\left(a-60\right)\hat{i} -55 \hat{j} \right\}$
On comparing, we get
$-20 = \left(a - 60\right)\lambda$ and $-11 = -55\lambda$
$\Rightarrow \lambda = 1/5$
$\therefore -20 = \left(a - 60\right)\left(1/5\right)$
$\Rightarrow a - 60 = -100$
$\Rightarrow a = -100 + 60$
$\Rightarrow a = -40$