Q. Let $a,b$ and $c$ be distinct non-negative numbers. If the vectors $a\widehat{\dot{\mathbf{i}}} + a\widehat{\dot{\mathbf{j}}} + c\widehat{\mathbf{k}},\ \widehat{\dot{\mathbf{i}}} + \widehat{\mathbf{k}}$ and $c\widehat{\dot{\mathbf{i}}} + c\widehat{\dot{\mathbf{j}}} + b\widehat{\mathbf{k}}$ lie in a plane, then $c$ is
A) The harmonic mean of $a$ and $b$
B) Equal to zero
C) The arithmetic mean of $a$ and $b$
D) The geometric mean of $a$ and $b$
Explanation
(d)
Since the given points lie in a plane.
$\therefore\ \left| \begin{matrix}
a & a & c \\
1 & 0 & 1 \\
c & c & b
\end{matrix} \right| = 0$
$Applying\ \ C_{1} \rightarrow C_{1} - C_{2}$
$\Longrightarrow \ \left| \begin{matrix}
0 & a & c \\
1 & 0 & 1 \\
0 & c & b
\end{matrix} \right| = 0$
$\Longrightarrow - 1\left( ab - c^{2} \right) = 0$
⟹ $c^{2} = ab$
Hence, $c$ is GM of $a\ and\ b$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.