Question

Consider the following statements:
% Statement Statement (I): If either \( |\vec{a}| = 0 \) or \( |\vec{b}| = 0 \), then \( \vec{a} \cdot \vec{b} = 0 \).
% Statement Statement (II): If \( \vec{a} \times \vec{b} = 0 \), then \( \vec{a} \) is perpendicular to \( \vec{b} \).
Which of the following is correct?

• A. Statement (I) is false but Statement (II) is true
• B. Both Statement (I) and Statement (II) are true
• C. Both Statement (I) and Statement (II) are false
• D. Statement (I) is true but Statement (II) is false

Hint:

In vector analysis, the cross product \( \vec{a} \times \vec{b} = 0 \) implies that the vectors are parallel (not perpendicular), while the dot product \( \vec{a} \cdot \vec{b} = 0 \) implies perpendicular vectors.

Answer 1

The Correct Option is D

Statement (I): If \( \vec{a} \cdot \vec{b} = 0 \), then \( |\vec{a}| = 0 \) or \( |\vec{b}| = 0 \) or \( \theta = \dfrac{\pi}{2} \). This is correct.
Statement (II): If \( \vec{a} \times \vec{b} = 0 \), then \( |\vec{a}| = 0 \) or \( |\vec{b}| = 0 \) or \( \theta = 0 \) or \( \pi \). This is incorrect because \( \vec{a} \) is parallel to \( \vec{b} \) when the cross product is zero, not perpendicular.
Thus, Statement (I) is true and Statement (II) is false.