Question

If \( \vec{a} \) is a non-zero vector then \( |\vec{a} \times \vec{a}| \) is equal to:

• A. \( |\vec{a}| \)
• B. \( |\vec{a}|^2 \)
• C. 1
• D. 0

Hint:

Remember: \( \vec{a} \cdot \vec{a} = |\vec{a}|^2 \) (Dot product) but \( \vec{a} \times \vec{a} = \vec{0} \) (Cross product).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The cross product of two vectors \( \vec{u} \) and \( \vec{v} \) depends on the sine of the angle \( \theta \) between them.
Step 2: Detailed Explanation:
The magnitude of the cross product is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \] In this problem, both vectors are the same (\( \vec{a} \)).
The angle between a vector and itself is \( \theta = 0^\circ \).
Since \( \sin 0^\circ = 0 \):
\[ |\vec{a} \times \vec{a}| = |\vec{a}| |\vec{a}| \sin 0^\circ = 0 \] Thus, the cross product of a vector with itself is always the null vector, and its magnitude is 0.
Step 3: Final Answer:
The value is 0.