1. ## dot products

Simplify ||x+y||^2 -||x||^2-||y||^2 using dot products

2. $\displaystyle <x+y,x+y>=<x,x+y>+<y,x+y>=<x,x>+<x,y>\cdots$

3. Continuing

$\displaystyle <x+y,x+y>=||x||^2+2<x,y>+||y||^2$

4. Originally Posted by Taurus3
Simplify ||x+y||^2 -||x||^2-||y||^2 using dot products
I wish helpers knew what is needed.
$\displaystyle \left\| {x + y} \right\|^2 = \left( {x + y} \right) \cdot \left( {x + y} \right) = x \cdot x + 2x \cdot y + y \cdot y =$
$\displaystyle \left\| x \right\|^2 + 2x \cdot y + \left\| y \right\|^2$

5. Originally Posted by Plato
I whish helpers knew what is needed.
$\displaystyle \left\| {x + y} \right\|^2 = \left( {x + y} \right) \cdot \left( {x + y} \right) = x \cdot x + 2x \cdot y + y \cdot y =$
$\displaystyle \left\| x \right\|^2 + 2x \cdot y + \left\| y \right\|^2$
That makes no sense since I supplied that exact answer. Additionally, I gave him/her the opportunity to finish it themselves.

6. oh wow. That was pretty confusing haha.

7. Originally Posted by Taurus3
oh wow. That was pretty confusing haha.
Taurus3, all the math in the above posts are correct, and trying to help you reach the answer.
What parts are you confused about?

As a side note, you can also find the value of ||x+y||^2 -||x||^2-||y||^2 using the law of cosines.

8. Originally Posted by dwsmith
That makes no sense since I supplied that exact answer.
Did I miss something?

9. Your solutions are wrong since you didn't transpose.

The standard inner product for $\displaystyle \mathbb{R}$ is the scalar product

$\displaystyle <x,y>=x^T\cdot y$.

$\displaystyle x_{n,1}\cdot x_{n,1}=DNE \ \ \mbox{but} \ x_{1,n}\cdot x_{n,1}=\alpha \ \ \alpha\in\mathbb{R}$

You are only correct if the vectors are 1,1 but you can't make that assumption.

10. Originally Posted by dwsmith
Your solutions are wrong since you didn't transpose.

The standard inner product for $\displaystyle \mathbb{R}$ is the scalar product

$\displaystyle <x,y>=x^T\cdot y$.

$\displaystyle x_{n,1}\cdot x_{n,1}=DNE \ \ \mbox{but} \ x_{1,n}\cdot x_{n,1}=\alpha \ \ \alpha\in\mathbb{R}$

You are only correct if the vectors are 1,1 but you can't make that assumption.
Dot product is not a matrix product of a row and column vector (so no transpose required). Plato's approach is correct and it is the approach that is expected (Your inner product is not a "dot" product as required by the question. You are assuming more knowledge on the part of the OP and so what you post is probably gobbledygook to them)

CB