Question

Let \( \vec{a} \) be any vector such that \( |\vec{a}| = a \). The value of \[ |\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 \] is:

• A. \( a^2 \)
• B. \( 2a^2 \)
• C. \( 3a^2 \)
• D. \( 0 \)

Hint:

Cross product magnitudes depend on sine of the angle between vectors.

Answer 1

The Correct Option is B

Step 1: Recall the formula for cross product magnitudes
The magnitude of the cross product is: \[ |\vec{a} \times \hat{i}| = |\vec{a}||\hat{i}|\sin\theta. \]
Step 2: Evaluate each term
For \( \vec{a} \times \hat{i} \), \( \vec{a} \times \hat{j} \), and \( \vec{a} \times \hat{k} \), the contributions along two directions add up, giving: \[ |\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2 = 2a^2. \]
Step 3: Verify the options
The correct result is \( 2a^2 \), matching option (B).