Question

If \( \vec{a} = 5\hat{i} - 7\hat{j} + 9\hat{k} \) and \( \vec{b} = -5\hat{i} + 7\hat{j} - 9\hat{k} \), then \( \vec{a} \cdot (\vec{a} \times \vec{b}) + (\vec{a} + \vec{b}) \cdot \hat{b} \) is equal to

• A. \( 50 \)
• B. \( -50 \)
• C. \( 49 \)
• D. \( -49 \)
• E. \( 0 \)

Hint:

The magnitude of the cross product of two vectors represents the area of the parallelogram they form.

Answer 1

Answer: (E)

Using vector properties: \[ \vec{a} \cdot (\vec{a} \times \vec{b}) = 0 \quad {(since triple scalar product is zero)} \] \[ (\vec{a} + \vec{b}) \cdot \hat{b} = 0 \] Thus, the final value is: \[ 0 \] So, the correct answer is (E).