Let a = {i} - {k}, b = x {i} + {j} + (1-x) {k} and c = y {i} + x {j} + (1+x-y) {k}. Then, [a b c] depends on

Correct Answer is : neither xxx nor yyy

Correct Answer is : neither xxx nor yyy

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BITSAT 2024 Shift 1 Paper

Question : 69 of 130

Marks: +1, -0

Let

a = \hat{i} - \hat{k}, b = x \hat{i} + \hat{j} + (1-x) \hat{k}

c = y \hat{i} + x \hat{j} + (1+x-y) \hat{k}

[a b c]

Solution:

To compute the scalar triple product

[a b c]

, we first use the determinant representation:

[a b c] = \begin{vmatrix} 1 & 0 & -1 \\ x & 1 & 1-x \\ y & x & 1+x-y \end{vmatrix}

This can be expanded as follows:

[a b c] = 1((1 \cdot (1+x-y)) - (x \cdot (1-x))) - 1 (x^2-y)

Breaking it down further:

= 1 \cdot (1+x-y-x+x^2) - (x^2-y)

Simplifying the expression:

\; = (1+x-y-x+x^2) - x^2 + y

\; = 1

The result 1 is independent of the variables

x

and

y

. Therefore, the value of the scalar triple product

[a b c]

does not depend on

x

y

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