Vectors 3 Question 8
8. If $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=x \hat{\mathbf{i}}+\hat{\mathbf{j}}+(1-x) \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=y \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(1+x-y) \hat{\mathbf{k}}$. Then, $[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$ depends on
(2001, 2M)
(a) only $x$
(b) only $y$
(c) Neither $x$ nor $y$
(d) both $x$ and $y$
Show Answer
Answer:
Correct Answer: 8. (c)
Solution:
- $[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]=\left|\begin{array}{ccc}1 & 0 & -1 \ x & 1 & 1-x \ y & x & 1+x-y\end{array}\right|$ Applying $C _3 \rightarrow C _1+C _3$,
$$ \left|\begin{array}{ccc} 1 & 0 & 0 \\ x & 1 & 1 \\ y & x & 1+x \end{array}\right|=1 $$
Therefore, it neither depends on $x$ nor $y$.
