# If IA+BI = IA-BI, find the angle between the vector A and B and show that the two vectos are perpendicular to each other. How to answer this question?

##### 3 Answers

Let the angle between two vectors

So

Again

By the given condition

So

See below.

#### Explanation:

Considering that

If

or simplifying

This means that the scalar product of

We can use some properties of the Vector Norm.

# ||A|| = sqrt(A * A) => ||A||^2 = A * A #

We are given that:

# || A+B || = || A - B|| #

If we square both sides we get:

# || A+B ||^2 = || A - B||^2 #

Using the above property this becomes:

# (A+B) * (A+B) = (A - B) * (A-B) #

And the Vector Dot Product is distributive over vector addition so we can expand the above expression;

# (A+B) * A + (A+B) * B = (A - B) * A - (A - B) * B #

# :. A*A+B*A + A*B+B* B = A*A - B* A - A*B + B * B #

# :. B*A + A*B = - B* A - A*B #

And the vector product is Commutative so

# :. A*B + A*B = - A* B - A*B #

# :. 2A*B = -2 A* B #

# :. 4A*B = 0 #

# :. A*B = 0 #

And If

#A=0# or#B=0# ; or

#A# and#B# are perpendicular. QED