Question

Assertion (A): Projection of \( \vec{a} \) on \( \vec{b} \) is the same as the projection of \( \vec{b} \) on \( \vec{a} \).
Reason (R): The angle between \( \vec{a} \) and \( \vec{b} \) is the same as the angle between \( \vec{b} \) and \( \vec{a} \) numerically.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is {not} the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Projections depend on the norms of the vectors, while angles between vectors are independent of their order.

Answer 1

The Correct Option is D

The projection of \( \vec{a} \) on \( \vec{b} \) is: \[ {Proj}_{\vec{b}} (\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b}. \] The projection of \( \vec{b} \) on \( \vec{a} \) is: \[ {Proj}_{\vec{a}} (\vec{b}) = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|^2} \vec{a}. \] Clearly, the two projections are not the same unless \( \|\vec{a}\| = \|\vec{b}\| \). The angle between \( \vec{a} \) and \( \vec{b} \) is the same as the angle between \( \vec{b} \) and \( \vec{a} \), as the cosine function is symmetric. Thus, the reason is true.
Final Answer: \( \boxed{{[(D)] Assertion (A) is false, but Reason (R) is true.}} \)