Question

Assertion (A): For any non-zero unit vector \( \vec{a} \), \( \vec{a} \cdot (-\vec{a}) = (-\vec{a}) \cdot \vec{a} = -1 \). 
Reason (R): Angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \). 
 

• 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.
• 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.

Answer 1

The Correct Option is C

Step 1: Validate Assertion (A)
The dot product of a vector \( \vec{a} \) and its negation \( -\vec{a} \) is defined as: \[ \vec{a} \cdot (-\vec{a}) = |\vec{a}| \cdot |-\vec{a}| \cdot \cos \theta. \] Given that \( \vec{a} \) is a unit vector, \( |\vec{a}| = 1 \). Consequently, \( |-\vec{a}| = 1 \) as well. The angle \( \theta \) between \( \vec{a} \) and \( -\vec{a} \) is \( \pi \) (180°), for which \( \cos \pi = -1 \). Substituting these values, we get: \[ \vec{a} \cdot (-\vec{a}) = 1 \cdot 1 \cdot (-1) = -1. \]
Thus, Assertion (A) is confirmed as true. 
Step 2: Validate Reason (R)
Reason (R) asserts that the angle between \( \vec{a} \) and \( -\vec{a} \) is \( \frac{\pi}{2} \) (90°). This is incorrect. As \( \vec{a} \) and \( -\vec{a} \) are vectors pointing in opposite directions, the actual angle between them is \( \pi \) (180°). 
Therefore, Reason (R) is determined to be false. 
Conclusion: Assertion (A) is true, whereas Reason (R) is false. 
 

Answer 2

Step 1: Evaluate Assertion (A)
A scalar matrix is defined as a diagonal matrix with all diagonal elements equal and all non-diagonal elements zero.
Thus, Assertion (A) is accurate.
Step 2: Evaluate Reason (R)
A diagonal matrix allows any value for its diagonal elements; they are not restricted to zero. 
Consequently, Reason (R) is incorrect. 
Step 3: Final Determination
Assertion (A) is true, while Reason (R) is false. 
Therefore, option (C) is the correct choice.