The vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{j}+\hat{k}$ and bisects the angle between $\vec{b}$ and $\vec{c}$. Then which one of the following gives possible values of $\alpha$ and $\beta$ ?

Question

Aicharya · Accepted Answer

$ \begin{aligned} & \overrightarrow{\mathrm{a}}=\lambda(\hat{\mathrm{b}}+\hat{\mathrm{c}}) \ & \Rightarrow \alpha \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}=\lambda\left(\frac{\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{2}}ight) \ & \lambda=\sqrt{2} \alpha 	ext { and } \lambda=\sqrt{2} 	ext { and } \lambda=\sqrt{2} \beta \ & \Rightarrow \alpha=1 	ext { and } \beta=1 . \end{aligned} $

The vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}$ lies in the plane of the vectors…

Solution