Question

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

• A. only \( y \)
• B. only \( x \)
• C. both \( x \) and \( y \)
• D. neither \( x \) nor \( y \)

Hint:

When working with scalar triple products, simplify the determinant step by step and carefully analyze the dependence on variables.

Answer 1

The Correct Option is D

We are asked to find \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| \). 
Step 1: Compute the cross product \( \mathbf{b} \times \mathbf{c} \) and then dot it with \( \mathbf{a} \). \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \left| \begin{vmatrix} 1 & 0 & -1 \\ x & 1 & 1 - x \\ y & x & 1 + x - y \end{vmatrix} \right| \] Simplifying this determinant: \[ = 1 + x - y - x^2 + x^2 - y = 1 \] Thus, \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = 1 \), which is independent of both \( x \) and \( y \).