Question

The vectors $2\hat{i}+3\hat{j},5\hat{i}+6\hat{j}$ and $8\hat{i}+\alpha \hat{j}$ have their initial points at $(1,1)$. The value of $\alpha$ so that the vectors terminate on one straight line is

• A. $9$
• B. $3$
• C. $6$
• D. $0$

Answer 1

The correct option is A $9$


Since, initial point of $2\hat{i}+3\hat{j},5\hat{i}+6\hat{j}$ and $8\hat{i}+\alpha \hat{j}$ is $\hat{i}+\hat{j},$ hence their terminal points will be $3\hat{i}+4\hat{j},6\hat{i}+7\hat{j}$ and $9\hat{i}+(\alpha +1)\hat{j}.$

$\Rightarrow \overrightarrow{BC}=3\hat{i}+3\hat{j}$
and $\overrightarrow{BD}=6\hat{i}+(\alpha -3)\hat{j}$

Since, all the vectors terminate on one straight line, hence $B,C,D$ are collinear.
$\therefore \overrightarrow{BD}=k\overrightarrow{BC}\Rightarrow k(3\hat{i}+3\hat{j})=6\hat{i}+(\alpha -3)\hat{j}]\Rightarrow k=2;\alpha =9$