# Prove using general properties of vectors (not coordinates):

## Homework Statement

Prove that |X+Y|^2 - |X-Y|^2 = 4X(dot)Y using general properties of vectors.

???

## The Attempt at a Solution

I'm very confused about how to start this. If someone could give me some help to just get me started then that would hopefully get the ball rolling for me. Thanks!

Char. Limit
Gold Member
What's the definition of the magnitude squared? Might be a good way to start.

Homework Helper
Well, simply "expand" the left side of the equation...

Edit: sorry, late

Oh wow, I definitely wasn't thinking straight. Thanks guys, I expanded it and got 4XY. Well that was a lot easier than I thought...

SammyS
Staff Emeritus
Homework Helper
Gold Member
Oh wow, I definitely wasn't thinking straight. Thanks guys, I expanded it and got 4XY. Well that was a lot easier than I thought...
Did you get 4X(dot)Y ? ... or simply 4XY ?

I got just 4XY, but I don't see how I can get 4X(dot)Y from that??? Is there a way to say that 4XY = 4X(dot)Y? Thanks again for the help.

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Prove that |X+Y|^2 - |X-Y|^2 = 4X(dot)Y using general properties of vectors.

???

## The Attempt at a Solution

I'm very confused about how to start this. If someone could give me some help to just get me started then that would hopefully get the ball rolling for me. Thanks!

I got just 4XY, but I don't see how I can get 4X(dot)Y from that??? Is there a way to say that 4XY = 4X(dot)Y? Thanks again for the help.
I assume that X & Y are vectors, not variables used as coordinates.

So you need to :

$$\text{Prove that }\ \left|\vec{X}+\vec{Y}\right|^2-\left|\vec{X}-\vec{Y}\right|^2=4\vec{X}\cdot\vec{Y}\ \text{ using general properties of vectors.}$$

Some properties:
$$\left|\vec{A}\,\right|^2=\vec{A}\cdot\vec{A}$$

$$\vec{A}\cdot\left(\vec{B}+\vec{C}\right)=\vec{A}\cdot\vec{B}+\vec{A}\cdot\vec{C}\quad\text{and}\quad\left(\vec{B}+\vec{C}\right)\cdot\vec{A}=\vec{B}\cdot\vec{A}+\vec{C}\cdot\vec{A}$$

$$\vec{A}\cdot\vec{B}=\vec{B}\cdot\vec{A}$$

etc.​
For instance:

$$\left|\vec{X}+\vec{Y}\right|^2=\vec{X}\cdot\vec{X}+2\vec{X}\cdot\vec{Y}+\vec{Y}\cdot\vec{Y}$$