Q. The vector $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + m\widehat{\mathbf{k}}$, $\overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + (m + 1)\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + m\widehat{\mathbf{k}}$ are coplanar, if $m$ is equal to
A) 1
B) 4
C) 3
D) No value of $m$ for which vectors are coplanar
Explanation
(d)
Given vectors are coplanar, if
$\left| \begin{matrix}
1 & 1 & m \\
1 & 1 & m + 1 \\
1 & - 1 & m
\end{matrix} \right| = 0$
⟹$\left| \begin{matrix}
0 & 0 & - 1 \\
1 & 1 & m + 1 \\
1 & - 1 & m
\end{matrix} \right| = 0\ \ \lbrack R_{1} \rightarrow R_{1} - R_{2}\rbrack$
⟹ $- 1( - 1 - 1) = 0$
$\Longrightarrow 2 \neq 0$
$\therefore$ Now value of $m$ for which vectors are coplanar.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.