Question

If $|A+B| = |A-B|$ then the angle between the two vectors $A$ and $B$ is

• A. $0^{\circ}$
• B. $180^{\circ}$
• C. $120^{\circ}$
• D. $90^{\circ}$

Hint:

Geometrically, this means the diagonals of a parallelogram are equal, which only happens if the parallelogram is a rectangle (angle = $90^{\circ}$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The magnitude of the sum and difference of two vectors depends on the angle \( \theta \) between them. We use the parallelogram law of vector addition to set up an equation.
Step 2: Key Formula or Approach:
1. \( |\vec{A} + \vec{B}|^2 = A^2 + B^2 + 2AB \cos \theta \).
2. \( |\vec{A} - \vec{B}|^2 = A^2 + B^2 - 2AB \cos \theta \).
Step 3: Detailed Explanation:
Square both sides of the given condition: \[ |\vec{A} + \vec{B}|^2 = |\vec{A} - \vec{B}|^2 \] \[ A^2 + B^2 + 2AB \cos \theta = A^2 + B^2 - 2AB \cos \theta \] Cancel \( A^2 \) and \( B^2 \) from both sides: \[ 2AB \cos \theta = -2AB \cos \theta \] \[ 4AB \cos \theta = 0 \] Since \( A \) and \( B \) are non-zero vectors: \[ \cos \theta = 0 \implies \theta = 90^\circ \]
Step 4: Final Answer:
The angle between the vectors is 90°.