Question

If $\vec A + \vec B + \vec C = 0$, then $\vec A \times \vec B$ is equal to:

• A. $\vec B \times \vec C$
• B. $\vec C \times \vec B$
• C. $\vec A \times \vec C$
• D. None of these

Hint:

Important vector identities:
$\vec A \times \vec A = 0$
$\vec A \times \vec B = -(\vec B \times \vec A)$
Cross product is distributive over addition These simplify many vector problems quickly.

Answer 1

The Correct Option is A

Step 1: Use the given vector relation. \[ \vec A + \vec B + \vec C = 0 \] \[ \Rightarrow \vec A = -(\vec B + \vec C) \]
Step 2: Take cross product with $\vec B$. \[ \vec A \times \vec B = -(\vec B + \vec C)\times \vec B \]
Step 3: Apply distributive property of cross product. \[ \vec A \times \vec B = -(\vec B \times \vec B + \vec C \times \vec B) \]
Step 4: Use properties of cross product. \[ \vec B \times \vec B = 0 \] So, \[ \vec A \times \vec B = -(\vec C \times \vec B) \]
Step 5: Use anti-commutative property. \[ \vec C \times \vec B = -(\vec B \times \vec C) \] Hence, \[ \vec A \times \vec B = \vec B \times \vec C \] Final Answer: \[ \boxed{\vec A \times \vec B = \vec B \times \vec C} \]