Question

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

• A. Neither \( x \) nor \( y \)
• B. Only \( x \)
• C. Only \( y \)
• D. Both \( x \) and \( y \)

Hint:

If the cross product of two vectors results in a zero vector, the scalar triple product will always be zero, independent of other variables.

Answer 1

The Correct Option is A

We are given the vectors: \[ \mathbf{a} = \hat{i} - \hat{k}, \mathbf{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}, \mathbf{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k} \] The scalar triple product \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] \) is given by: \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \] We will calculate \( \mathbf{b} \times \mathbf{c} \) and check if the result depends on \( x \) or \( y \).
Step 1: Calculate \( \mathbf{b} \times \mathbf{c} \) The cross product \( \mathbf{b} \times \mathbf{c} \) is: \[ \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x & 1 & 1 - x \\ y & x & 1 + x - y \end{vmatrix} \] Expanding the determinant: \[ \mathbf{b} \times \mathbf{c} = \hat{i} \left( \begin{vmatrix} 1 & 1 - x \\ x & 1 + x - y \end{vmatrix} \right) - \hat{j} \left( \begin{vmatrix} x & 1 - x \\ y & 1 + x - y \end{vmatrix} \right) + \hat{k} \left( \begin{vmatrix} x & 1 \\ y & x \end{vmatrix} \right) \] Upon solving the determinant, we get: \[ \mathbf{b} \times \mathbf{c} = \hat{i}(0) - \hat{j}(0) + \hat{k}(0) \] Thus, \[ \mathbf{b} \times \mathbf{c} = \mathbf{0} \]
Step 2: Compute the Scalar Triple Product Now that we know \( \mathbf{b} \times \mathbf{c} = \mathbf{0} \), we calculate the scalar triple product: \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \mathbf{a} \cdot \mathbf{0} = 0 \] Thus, the scalar triple product does not depend on either \( x \) or \( y \), so the answer is \( \boxed{\text{(a)}} \).