Question

If \(|\vec A+\vec B|=|\vec A-\vec B|\), then the angle between the two vectors \(\vec A\) and \(\vec B\) is

• A. \(0^\circ\)
• B. \(180^\circ\)
• C. \(120^\circ\)
• D. \(90^\circ\)

Hint:

If \(|\vec A+\vec B|=|\vec A-\vec B|\), then the vectors are perpendicular.

Answer 1

The Correct Option is D

Concept: For two vectors: \[ |\vec A+\vec B|^2=A^2+B^2+2AB\cos\theta \] and \[ |\vec A-\vec B|^2=A^2+B^2-2AB\cos\theta \]

Step 1: Given: \[ |\vec A+\vec B|=|\vec A-\vec B| \] Squaring both sides: \[ |\vec A+\vec B|^2=|\vec A-\vec B|^2 \]

Step 2: Substitute formulas. \[ A^2+B^2+2AB\cos\theta=A^2+B^2-2AB\cos\theta \]

Step 3: Cancel common terms. \[ 2AB\cos\theta=-2AB\cos\theta \] \[ 4AB\cos\theta=0 \] Since \(A\) and \(B\) are non-zero: \[ \cos\theta=0 \]

Step 4: Therefore: \[ \theta=90^\circ \] Hence, \[ \boxed{90^\circ} \]