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 (A) and (R) are true, and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true, but (R) is not the correct explanation of (A).
• C. (A) is true, but (R) is false.
• D. (A) is false, but (R) is true.
  %Correct answer \textbf{Correct answer:}\textbf{{Both (A) and (R) are true, but (R) is not the correct explanation of (A).}}

Hint:

The dot product is commutative, while the cross product is anti-commutative. Verify these properties independently.

Answer 1

The Correct Option is A

Step 1: Verify the Assertion (A).
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta, \] where \( \theta \) is the angle between the vectors. Since multiplication is commutative, \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \). Hence, (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 \( \vec{b} \). Reversing the order of the cross product: \[ \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}). \] Thus, \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \). Hence, (R) is true. Step 3: Relationship between (A) and (R).
The Reason (R) refers to the property of the cross product, while the Assertion (A) refers to the property of the dot product. Although both statements are true, (R) does not explain (A). Step 4: Conclusion.
Both (A) and (R) are true, but (R) is not the correct explanation of (A). Thus: {Both (A) and (R) are true, but (R) is not the correct explanation of (A).}