Question

A vector $\overrightarrow{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}(\alpha ,\beta \in R)$ lies in the plane of the vectors, $\overrightarrow{b}=\hat{i}+\hat{j}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\overrightarrow{a}$ bisects the angle between $\overrightarrow{b}$ and $\overrightarrow{c},$ then

• A. $\overrightarrow{a}.\hat{i}+3=0$
• B. $\overrightarrow{a}.\hat{k}+4=0$
• C. $\overrightarrow{a}.\hat{i}+1=0$
• D. $\overrightarrow{a}.\hat{k}+2=0$

Answer 1

The correct option is D $\overrightarrow{a}.\hat{k}+2=0$

NOTE : This is a BONUS question as none of the options satisfy the given condition.
The angle bisector of vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ is given by :
$\overrightarrow{a}=\lambda (\hat{b}+\hat{c})=\lambda (\frac{\hat{i}+\hat{j}}{\sqrt{2}}+\frac{\hat{i}-\hat{j}+4\hat{k}}{3\sqrt{2}})=\lambda \left(\frac{4\hat{i}+2\hat{j}+4\hat{k}}{3}\right)$
Comparing with $\overrightarrow{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k},$ we get
$\frac{2\lambda }{3\sqrt{2}}=2\Rightarrow \lambda =3\sqrt{2}$
$\therefore \overrightarrow{a}=4\hat{i}+2\hat{j}+4\hat{k}$
None of the options satisfy.