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 \( y \). 
 

Answer 2

Step 1: Understanding the Given Vectors
We are provided with three 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 goal is to find the value of the scalar triple product \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] \), which is given by the determinant of the matrix formed by the components of the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \).
Step 2: Set up the Determinant
The scalar triple product is defined as: \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & -1 \\ x & 1 & 1 - x \\ y & x & 1 + x - y \end{vmatrix} \]
Step 3: Calculate the Determinant
We calculate the determinant of the 3x3 matrix: \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \hat{i} \begin{vmatrix} 0 & -1 \\ 1 & 1 - x \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & -1 \\ y & 1 + x - y \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 0 \\ y & x \end{vmatrix} \] We compute the individual 2x2 determinants: 1. For \( \hat{i} \)-component: \[ \begin{vmatrix} 0 & -1 \\ 1 & 1 - x \end{vmatrix} = (0)(1 - x) - (-1)(1) = 1 \] 2. For \( \hat{j} \)-component: \[ \begin{vmatrix} 1 & -1 \\ y & 1 + x - y \end{vmatrix} = (1)(1 + x - y) - (-1)(y) = 1 + x - y + y = 1 + x \] 3. For \( \hat{k} \)-component: \[ \begin{vmatrix} 1 & 0 \\ y & x \end{vmatrix} = (1)(x) - (0)(y) = x \] So, the scalar triple product becomes: \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \hat{i}(1) - \hat{j}(1 + x) + \hat{k}(x) \] Simplifying: \[ [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \hat{i} - (1 + x)\hat{j} + x\hat{k} \]
Step 4: Analyze the Dependence on \( x \) and \( y \)
Notice that the result contains no terms that depend on \( y \). Furthermore, the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) do not involve \( x \) in such a way that the scalar triple product itself becomes zero for any non-trivial values of \( x \) and \( y \). Therefore, the scalar triple product does not depend on either \( x \) or \( y \).
Step 5: Conclusion
Thus, the scalar triple product \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] \) does not depend on \( x \) or \( y \). The correct answer is: \[ \boxed{\text{neither } x \text{ nor } y} \]