Q. Let $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} - \widehat{\mathbf{k}},\ \overrightarrow{\mathbf{b}} = x\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + (1 - x)\widehat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}} = y\widehat{\dot{\mathbf{i}}} + x\widehat{\dot{\mathbf{j}}} + (1 + x - y)\widehat{\mathbf{k}}.$ Then $\lbrack\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}},\ \overrightarrow{\mathbf{c}}\rbrack$ depends on
A) neither $x$ nor $y$
B) both $x$ and $y$
C) only $x$
D) only $y$
Explanation
(a)
$\left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack = \left| \begin{matrix}
1 & 0 & - 1 \\
x & 1 & 1 - x \\
y & x & 1 + x - y
\end{matrix} \right|$
Applying $C_{3} \rightarrow C_{3} + C_{1}$
$= \left| \begin{matrix}
1 & 0 & 0 \\
x & 1 & 1 \\
y & x & 1 + x
\end{matrix} \right| = 1\lbrack 1 + x - x\rbrack = 1$
Hence, $\left\lbrack \overrightarrow{\mathbf{a}}\ \ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack$ does not depend upon neither $x$ nor $y$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.