Question

If \[ \mathbf{a} = 2\hat{i} + 3\hat{j} - \hat{k}, \quad \mathbf{b} = -\hat{i} + 2\hat{j} - 4\hat{k}, \quad \mathbf{c} = \hat{i} + \hat{j} + \hat{k} \] then \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c}) = \)

• A. -74
• B. 64
• C. -64
• D. 74

Hint:

To compute the dot product of cross products, first compute each cross product using the determinant and then take the dot product of the resulting vectors.

Answer 1

The Correct Option is A

Step 1: Find the cross products.
To solve for \( (\mathbf{a} \times \mathbf{b}) \) and \( (\mathbf{a} \times \mathbf{c}) \), compute the cross products: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ -1 & 2 & -4 \end{vmatrix} \] \[ \mathbf{a} \times \mathbf{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1 \end{vmatrix} \]
Step 2: Take the dot product.
Now, take the dot product of the two resulting vectors: \[ (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c}) = -74 \]
Step 3: Conclusion.
Thus, the value is -74, corresponding to option (A).