Vectors proof question

• Mar 13th 2013, 05:20 PM
Vishak
Vectors proof question
Hi,can someone please explain how to solve this question to me.

1) For any two vectors u and v, prove that |u + v|^2 + |u -v|^2 = 2(|u|^2 + |v|^2)

In proving this, what geometric fact have your proved?

Thanks heaps!!
• Mar 13th 2013, 05:26 PM
ILikeSerena
Re: Vectors proof question
Quote:

Originally Posted by Vishak
Hi,can someone please explain how to solve this question to me.

1) For any two vectors u and v, prove that |u + v|^2 + |u -v|^2 = 2(|u|^2 + |v|^2)

In proving this, what geometric fact have your proved?

Thanks heaps!!

Hi Vishak! :)

Note that $\displaystyle |u+v|^2 = \langle u+v, u+v \rangle$.
Can you simplify that?
• Mar 13th 2013, 05:45 PM
Vishak
Re: Vectors proof question
I'm not sure what the triangle brackets mean, but ill guess - is it just u^2 + 2uv + v^2?
• Mar 13th 2013, 05:45 PM
Plato
Re: Vectors proof question
Quote:

Originally Posted by Vishak
1) For any two vectors u and v, prove that |u + v|^2 + |u -v|^2 = 2(|u|^2 + |v|^2)
In proving this, what geometric fact have your proved?

Surely you must know that $\displaystyle \|u+v\|^2=(u+v)\cdot(u+v)~\&~\|u-v\|^2=(u-v)\cdot(u-v)~!$

Moreover, if vectors $\displaystyle u~\&~v$ are adjacent sides of a parallelogram then $\displaystyle (u+v)~\&~(u-v)$ are its diagonals.
• Mar 13th 2013, 10:20 PM
Vishak
Re: Vectors proof question
Thanks guys, what about the second part - "what geometric fact have you proved?"
• Mar 14th 2013, 02:51 AM
ILikeSerena
Re: Vectors proof question
Quote:

Originally Posted by Vishak
I'm not sure what the triangle brackets mean, but ill guess - is it just u^2 + 2uv + v^2?

The triangle brackets are one of the ways you can write a dot product of vectors.
You can also write it as $\displaystyle (u+v) \cdot (u+v)$.

And yes, that is what it is.

Quote:

Originally Posted by Vishak
Thanks guys, what about the second part - "what geometric fact have you proved?"

Care to guess now that you know that $\displaystyle u~\&~v$ are adjacent sides of a parallelogram and $\displaystyle (u+v)~\&~(u-v)$ are its diagonals?