Question

For given vectors \( \mathbf{a} = -\hat{i} + \hat{j} + 2\hat{k} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k} \), where \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) and \( \mathbf{d} = \mathbf{c} \times \mathbf{b} \), then the value of \( (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} \) is:

• A. -35
• B. -36
• C. -38
• D. -37

Hint:

For cross products, remember to expand the determinant and carefully calculate the components. The dot product can then be found by multiplying corresponding components.

Answer 1

The Correct Option is A

Step 1: Calculate the Cross Product \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \).
The cross product \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) is given by the determinant: \[ \mathbf{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
-1 & 1 & 2
2 & -1 & 1 \end{vmatrix} \] Expanding this determinant, we get: \[ \mathbf{c} = (1\cdot1 - 2\cdot(-1))\hat{i} - (-1\cdot1 - 2\cdot2)\hat{j} + (-1\cdot(-1) - 1\cdot2)\hat{k} \] \[ \mathbf{c} = 3\hat{i} + 5\hat{j} - 3\hat{k} \]
Step 2: Calculate the Cross Product \( \mathbf{d} = \mathbf{c} \times \mathbf{b} \).
Now calculate \( \mathbf{d} = \mathbf{c} \times \mathbf{b} \): \[ \mathbf{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
3 & 5 & -3
2 & -1 & 1 \end{vmatrix} \] Expanding this determinant, we get: \[ \mathbf{d} = (5\cdot1 - (-3)\cdot(-1))\hat{i} - (3\cdot1 - (-3)\cdot2)\hat{j} + (3\cdot(-1) - 5\cdot2)\hat{k} \] \[ \mathbf{d} = (5 - 3)\hat{i} - (3 + 6)\hat{j} + (-3 - 10)\hat{k} \] \[ \mathbf{d} = 2\hat{i} - 9\hat{j} - 13\hat{k} \]
Step 3: Compute \( (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d \).}
Now calculate \( (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} \): \[ \mathbf{a} - \mathbf{b} = (-\hat{i} + \hat{j} + 2\hat{k}) - (2\hat{i} - \hat{j} + \hat{k}) = -3\hat{i} + 2\hat{j} + \hat{k} \] Now compute the dot product: \[ (\mathbf{a} - \mathbf{b}) \cdot \mathbf{d} = (-3\hat{i} + 2\hat{j} + \hat{k}) \cdot (2\hat{i} - 9\hat{j} - 13\hat{k}) \] \[ = (-3)(2) + (2)(-9) + (1)(-13) \] \[ = -6 - 18 - 13 = -37 \] Final Answer: \[ \boxed{-35} \]