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: Try numbers:
Take $\vec A = 2\hat i$ and $\vec B = 3\hat j$. Then $\vec A + \vec B = 2\hat i + 3\hat j = \vec B + \vec A$.

Step 2: Compare the others:
$\vec A\cdot\vec B = 0$ either way, $\vec A\times\vec B = 6\hat k$ while $\vec B\times\vec A = -6\hat k$, and $\vec A - \vec B = 2\hat i - 3\hat j$ differs from $\vec B - \vec A = -2\hat i + 3\hat j$. Only (C) is an identity.

Final Answer:
Option (C). \[ \boxed{\text{(C)}} \]