Prove using general properties of vectors (not coordinates):

[ccmetz2020]

Homework Statement

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

Homework Equations

?

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]

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

 

[radou]

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

Edit: sorry, late

 

[ccmetz2020]

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]

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 ?

 

[ccmetz2020]

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]

Homework Statement

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

Homework Equations

?

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:

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