SATHEE: Vectors 3 Question 8

Vectors 3 Question 8

8. If $\vec{a} = \hat{i} - \hat{k}, \vec{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\vec{c} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ . Then, $[\vec{a} \vec{b} \vec{c}]$ 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:

$[\vec{a} \vec{b} \vec{c}] = | \begin{array}{ccc} 1 & 0 & - 1 x & 1 & 1 - x y & x & 1 + x - y \end{array} |$ Applying $C_{3} \to C_{1} + C_{3}$ ,

$| \begin{array}{ccc} 1 & 0 & 0 \\ x & 1 & 1 \\ y & x & 1 + x \end{array} | = 1$

Therefore, it neither depends on $x$ nor $y$ .

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