Question

Assertion (A): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).

• 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 \textit{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. %Correct answer \textbf{Correct answer:}\textbf{{ Assertion (A) is true, but Reason (R) is false.}}

Hint:

Remember, the dot product is commutative, while the cross product is anti-commutative. These are distinct properties, so verify each separately when comparing.

Answer 1

The Correct Option is C

Step 1: Verify the Assertion (A).
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is defined as: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta, \] where \( \theta \) is the angle between the vectors. The dot product is commutative, which means that: \[ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}. \] Thus, Assertion (A) is true. Step 2: Verify the Reason (R).
The cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \, \hat{n}, \] where \( \hat{n} \) is a unit vector perpendicular to both \( \vec{a} \) and \( \vec{b} \). The cross product is anti-commutative, which means: \[ \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}). \] This statement is valid for the cross product, but it does not relate to the commutative property of the dot product. Hence, Reason (R) is false when used as an explanation for (A). Step 3: Compare (A) and (R).
The Assertion (A) is correct because the dot product is commutative. However, Reason (R) incorrectly discusses the anti-commutative property of the cross product, which is unrelated to the Assertion. Therefore: \[ \text{A is true, but R is false.} \]