Question

For three non-zero vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$, $\vec{A} + \vec{B} = \vec{C}$ and $A^2 + B^2 = C^2$, then the angle between $\vec{A}$ and $\vec{B}$ will be

• A. $90^\circ$
• B. $180^\circ$
• C. $30^\circ$
• D. $60^\circ$

Hint:

If $A^2 + B^2 = C^2$ for $\vec{C} = \vec{A} + \vec{B}$, the vectors are perpendicular.

Answer 1

The Correct Option is A

Step 1: Use vector identity.
\[ C^2 = (\vec{A} + \vec{B})^2 = A^2 + B^2 + 2\vec{A}\cdot\vec{B} \] Step 2: Compare with given condition.
Given: \[ A^2 + B^2 = C^2 \] Step 3: Substitute and simplify.
\[ A^2 + B^2 = A^2 + B^2 + 2AB\cos\theta \] \[ 2AB\cos\theta = 0 \] Step 4: Find angle.
\[ \cos\theta = 0 \Rightarrow \theta = 90^\circ \]