Question

If \(\vec A+\vec B=\vec C\) and \(A^2+B^2=C^2\), then the angle between vectors \(\vec A\) and \(\vec B\) is

• A. \(0^\circ\)
• B. \(60^\circ\)
• C. \(90^\circ\)
• D. \(120^\circ\)

Hint:

If \(C^2=A^2+B^2\), then vectors \(\vec A\) and \(\vec B\) are perpendicular.

Answer 1

The Correct Option is C

Concept:
When two vectors \(\vec A\) and \(\vec B\) are added, the magnitude of resultant \(\vec C\) is given by: \[ C^2=A^2+B^2+2AB\cos\theta \] where \(\theta\) is the angle between \(\vec A\) and \(\vec B\).

Step 1: Given: \[ \vec A+\vec B=\vec C \]

Step 2: Also given: \[ A^2+B^2=C^2 \]

Step 3: From vector addition: \[ C^2=A^2+B^2+2AB\cos\theta \]

Step 4: Compare with: \[ C^2=A^2+B^2 \]

Step 5: Therefore: \[ 2AB\cos\theta=0 \]

Step 6: Since \(A\) and \(B\) are non-zero vectors: \[ \cos\theta=0 \] \[ \theta=90^\circ \] \[ \boxed{90^\circ} \]