Question

There are two vectors \(\overset{⃗}{A} = 6\hat{i}+9\hat{j}-\hat{k}\) and \(\overset{⃗}{B} = 2\hat{i}+3\hat{j}-p\hat{k}\) which have the same direction. The value of 'p' is

• A. \(3\)
• B. \(\frac{1}{3}\)
• C. \(\frac{2}{3}\)
• D. \(-\frac{1}{3}\)

Hint:

Addition and dot product are commutative; the cross product is anti-commutative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For any two vectors, vector addition and the scalar (dot) product are commutative, while the vector (cross) product changes sign when the order is reversed. The magnitudes being different makes no difference to these rules.

Step 2: Check (A):
\(\vec A\cdot\vec B = AB\cos\theta = \vec B\cdot\vec A\), not \(-\vec B\cdot\vec A\) (unless it is zero). So (A) is false.

Step 3: Check (B):
\(\vec A\times\vec B = -\vec B\times\vec A\). So (B) is false for non-parallel vectors.

Step 4: Check (C) and (D):
\(\vec A + \vec B = \vec B + \vec A\) always holds, so (C) is true. Subtraction is not commutative: \(\vec A - \vec B = -(\vec B - \vec A)\), so (D) is false.

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