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 $y$

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Answer:

Correct Answer: 8. (c)

Solution:

  1. $[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]=\left|\begin{array}{ccc}1 & 0 & -1 \ x & 1 & 1-x \ y & x & 1+x-y\end{array}\right|$ 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 $y$.



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