Question

For what angle between A and B, |A+B| = | A-B .

Answer 1

Answer:

A = |A|

B = |B|

|A + B| = √(A2 + B2 + 2ABcosθ)

|A - B| = √(A2 + B2 - 2ABcosθ)

|A + B| = |A - B| ⇒

√(A2 + B2 + 2ABcosθ) = √(A2 + B2 - 2ABcosθ)

Square both sides.

A2 + B2 + 2ABcosθ = A2 + B2 - 2ABcosθ

2ABcosθ = -2ABcosθ

If A ≠ 0 and B ≠ 0, then

cosθ = -cosθ ⇒

cosθ = 0 ⇒ θ = 90°

Answer 2

Given :

| A + B | = | A - B |

To find :

The angle between vector A and B .

Solution :

By vector addition , we know ,

| A + B | = √( A^2 + B^2 + 2 *A*B*Cosθ )

and ,  | A - B | = √( A^2 + B^2 - 2 *A*B*Cosθ )

now, given ,

| A + B | = | A - B |

=> √( A^2 + B^2 + 2 *A*B*Cosθ ) =  √( A^2 + B^2 - 2 *A*B*Cosθ )

squaring both sides ,

=>  A^2 + B^2 + 2 *A*B*Cosθ  =   A^2 + B^2 - 2 *A*B*Cosθ

=> Cosθ  = - Cosθ

=> 2 Cosθ  = 0

=> Cosθ  = 0

=> θ  = 90°  

The value of the angle between vector A and B is 90° .