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: Dot product
$AB\cos\theta$ does not change when the vectors swap, so it cannot equal its own negative.

Step 2: Cross product and difference
Swapping the order of a cross product or a subtraction flips the sign, so (B) and (D) fail for unequal vectors.

Step 3: Addition
Placing the vectors head to tail in either order gives the same resultant, so (C) holds.

Final Answer:
Option (C). \[ \boxed{\vec A + \vec B = \vec B + \vec A} \]