Question

Let \(\overset{⃗}{A}\) and \(\overset{⃗}{B}\) are two non-zero vectors of different magnitude. Which one of the following is the correct equation ?

• A. \(\overset{⃗}{A}\cdot \overset{⃗}{B} = -\overset{⃗}{B}\cdot \overset{⃗}{A}\)
• B. \(\overset{⃗}{A}\times \overset{⃗}{B} = \overset{⃗}{B}\times \overset{⃗}{A}\)
• C. \(\overset{⃗}{A}+\overset{⃗}{B} = \overset{⃗}{B}+\overset{⃗}{A}\)
• D. \(\overset{⃗}{A}-\overset{⃗}{B} = \overset{⃗}{B}-\overset{⃗}{A}\)

Hint:

Vector addition is commutative, but the dot product is symmetric and the cross product is antisymmetric.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Vector addition obeys \(\vec A + \vec B = \vec B + \vec A\). The dot product is commutative, and the cross product reverses sign when the order is swapped.

Step 2: Check option (A)
\(\vec A \cdot \vec B = AB\cos\theta = \vec B \cdot \vec A\). A minus sign is wrong, since both vectors are non-zero. So (A) is FALSE.

Step 3: Check option (B)
\(\vec A \times \vec B = -\vec B \times \vec A\), so the equality given is FALSE (unless they are parallel).

Step 4: Check option (C)
\(\vec A + \vec B = \vec B + \vec A\) is the commutative law of addition, which is TRUE.

Step 5: Check option (D)
\(\vec A - \vec B = -(\vec B - \vec A)\), so equality would need \(\vec A = \vec B\). They have different magnitudes, so (D) is FALSE.

Final Answer:
Only \(\vec A + \vec B = \vec B + \vec A\) is always correct, option (C). \[ \boxed{\vec A + \vec B = \vec B + \vec A} \]