Question

Assertion (A) : The vectors \( \vec{a} \) and \( (-2\vec{a}) \), where \( \vec{a} \neq \vec{0} \), are collinear vectors.
Reason (R) : \( \vec{a} \cdot (-2\vec{a}) = 0 \).

• 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:

Collinear vectors have a dot product of \( \pm 2|\vec{a}|^2 \), not zero. A zero dot product indicates orthogonal vectors, which are at an angle of \( 90^\circ \), making them perpendicular, not parallel/collinear.

Answer 1

The Correct Option is C

Concept:
• Collinear vectors are vectors that are parallel to the same line, meaning one can be expressed as a scalar multiple of the other (\( \vec{b} = \lambda \vec{a} \)).
• The dot product of two vectors is zero if and only if the vectors are perpendicular (orthogonal) to each other, or if at least one of them is a zero vector.

Step 1: Evaluate Assertion (A).
The two vectors given are \( \vec{a} \) and \( -2\vec{a} \). Let \( \vec{b} = -2\vec{a} \). Here, \( \vec{b} \) is a scalar multiple of \( \vec{a} \) with \( \lambda = -2 \). Since one vector is a scalar multiple of the other, they are parallel (acting along the same line in opposite directions), which makes them collinear. Therefore, Assertion (A) is true.

Step 2: Evaluate Reason (R).
Let us calculate the dot product given in the reason: \[ \vec{a} \cdot (-2\vec{a}) = -2(\vec{a} \cdot \vec{a}) = -2|\vec{a}|^2 \] Since it is given that \( \vec{a} \neq \vec{0} \), its magnitude \( |\vec{a}| > 0 \), meaning \( -2|\vec{a}|^2 \neq 0 \). Thus, the statement \( \vec{a} \cdot (-2\vec{a}) = 0 \) is completely false. Conclusion:
Assertion (A) is true, and Reason (R) is false. This matches option (C).