Question

If \( \vec{a} = \hat{i} + 5\hat{k}, \, \vec{b} = 2\hat{i} + 3\hat{k}, \, \vec{c} = 4\hat{i} - \hat{j} + 2\hat{k}, \, \vec{d} = \hat{i} - \hat{j} \), then \( (\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d}) = \)

• A. 12
• B. 20
• C. 30
• D. 10

Hint:

When calculating the dot and cross product, ensure to follow the determinant method for the cross product and apply the distributive property for the dot product.

Answer 1

The Correct Option is A

Step 1: Calculate the vector \( \vec{c} - \vec{a} \).
\[ \vec{c} - \vec{a} = (4\hat{i} - \hat{j} + 2\hat{k}) - (\hat{i} + 5\hat{k}) = 3\hat{i} - \hat{j} - 3\hat{k} \] Step 2: Calculate the cross product \( \vec{b} \times \vec{d} \).
\[ \vec{b} \times \vec{d} = \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2\hat{i} + 3\hat{k} & \hat{i} - \hat{j} \end{matrix} \right| = \hat{i} + 5\hat{j} - 3\hat{k} \] Step 3: Calculate the dot product.
Now calculate the dot product: \[ (\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d}) = (3\hat{i} - \hat{j} - 3\hat{k}) \cdot (\hat{i} + 5\hat{j} - 3\hat{k}) = 12 \] Step 4: Conclusion.
The correct answer is (A) 12.