Q. The vector $\overrightarrow{\mathbf{a}} = \alpha\widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + \beta\widehat{\mathbf{k}}$ lies in the plane of the vectors $\overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}}$ and $\overrightarrow{c} = \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$ and bisects the angle between $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ Then, which one of the following gives possible value of $\alpha$ and β?
A) α=1, β $= 1$
B) α=2, β $= 2$
C) $\alpha = 1,\ \beta = 2$
D) α=2, β $= 1$
Explanation
(a)
Since, $\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{c}}$ are coplanar.$\$
$\Longrightarrow \ \left| \begin{matrix}
\alpha & 2 & \beta \\
1 & 1 & 0 \\
0 & 1 & 1
\end{matrix} \right| = 0$
$\Longrightarrow \ \alpha(1 - 0) - 2(1 - 0) + \beta(1 - 0) = 0$
$\Longrightarrow \alpha + \beta = 2\$Which is possible for $\alpha = 1,\beta = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.