Question

If two vectors $ \vec{A} $ and $ \vec{B} $ are mutually perpendicular, then the component of $ \vec{A} \cdot \vec{B} $ along the direction of $ \vec{A} + \vec{B} $ is

• A. \( \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \)
• B. \( \sqrt{|\vec{A}|^2 - |\vec{B}|^2} \)
• C. \( \frac{|\vec{A}|^2 - |\vec{B}|^2}{\sqrt{|\vec{A}|^2 + |\vec{B}|^2}} \)
• D. \( \frac{|\vec{A}|^2 + |\vec{B}|^2}{\sqrt{|\vec{A}|^2 + |\vec{B}|^2}} \)

Hint:

When resolving a vector along another, use the dot product formula: \(\text{Component} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\). If vectors are perpendicular, their dot product is zero.

Answer 1

The Correct Option is C

We are given that \( \vec{A} \cdot \vec{B} \) is to be resolved along the direction of \( \vec{A} + \vec{B} \), and \( \vec{A} \perp \vec{B} \). That implies: \[ \vec{A} \cdot \vec{B} = 0 \] We compute the projection of \( \vec{A} - \vec{B} \) on the direction \( \vec{A} + \vec{B} \). The required component is: \[ \text{Component} = \frac{(\vec{A} - \vec{B}) \cdot (\vec{A} + \vec{B})}{|\vec{A} + \vec{B}|} \] Using the identity: \[ (\vec{A} - \vec{B}) \cdot (\vec{A} + \vec{B}) = |\vec{A}|^2 - |\vec{B}|^2 \] and \[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \] (because \( \vec{A} \cdot \vec{B} = 0 \), since they are perpendicular) So, \[ \text{Component} = \frac{|\vec{A}|^2 - |\vec{B}|^2}{\sqrt{|\vec{A}|^2 + |\vec{B}|^2}} \]