Question

If the points with position vectors $10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\mathrm{and}a\hat{i}+11\hat{j}$ are collinear, find the value of a.

Answer 1

Let $A,B,C$ be the points with position vectors $10\hat{i}+3\hat{j},12\hat{i}-5\hat{j},a\hat{i}+11\hat{j}$. Then,
$\overrightarrow{AB}=$Position vector of B $-$ Position vector of A
$$\begin{gathered} =12\hat{i}-5\hat{j}-10\hat{i}-3\hat{j} \\ =2\hat{i}-8\hat{j} \end{gathered}$$

$\overrightarrow{BC}=$Position vector of C $-$ Position vector of B
$$\begin{gathered} =a\hat{i}+11\hat{j}-12\hat{i}+5\hat{j} \\ =\left(a-12\right)\hat{i}+16\hat{j} \end{gathered}$$

Since, $A,B$ and $C$ are collinear.
∴ $\overrightarrow{AB}=\lambda \overrightarrow{BC}.$
$$\begin{gathered} \Rightarrow 2\hat{i}-8\hat{j}=\lambda \left(a-12\right)\hat{i}+\lambda 16\hat{j} \\ \Rightarrow 2=\lambda \left(a-12\right),-8=\lambda 16 \\ \Rightarrow 2=\lambda \left(a-12\right),\lambda =-\frac{1}{2} \\ \Rightarrow 2=-\frac{1}{2}\left(a-12\right) \\ \Rightarrow -a+12=4 \\ \Rightarrow a=8 \end{gathered}$$