Proving Vector A is Perpendicular to Vector B: Proving a 90° Angle

[mikee]

Homework Statement

Prove that Vector A is perpendicular to Vector B if |VectorA + VectorB| = |VectorA-VectorB| and use this to prove that the angle formed by joining any point on a circle to the end points of a diameter is 90 degrees,

Homework Equations

The Attempt at a Solution

I really have no idea where to begin, i no that VectorA (dot) VectorB=0 for them to be perpendicular but i still don't really understand the second part of the problem, if somebody could point me in the right direction that would be appreciated thank you

 

[Dick]

(A+B).(A+B)=A.A+2A.B+B.B. What's (A-B).(A-B)? How can they be equal? For the circle problem take A to be the vector connecting the center of the circle to your point and B to be the vector connecting the center to one of the endpoints of the diameter. Do you see it now?

 

[mikee]

For the first part i meant that if |A+B|=|A-B| then A and B are perpendicular and i forgot to note it is in 3 space A(Ai,Aj,Ak), and i see what your saying for the second part but as i am picturing it the angle will be any angle depending on the vectors? Like if the vectors are perpendicular of coarse the angle will be 90degrees but the way i am understanding the question is that no matter what the angle will be 90 degrees

 

[Dick]

It doesn't depend on how many dimensions the space has it's still true. If |A+B|=|A-B| then A.B=0. Use the vectors. |A+B|^2=(A+B).(A+B). |A-B|^2=(A-B).(A-B). Again for the circle problem A+B and A-B are both points on the circle.

 

[mikee]

oo ok i see now, the only way for part one of the question to hold true is if the vectors are perpendicular and by using these vectors the second part of the question would hold true, i just assumed that the second part of the question held true for any two vectors, thanks for the help