1. ## Orthogonality

Let $\displaystyle \vec{x}$ and $\displaystyle \vec{y}$ be linearly independent vectors in $\displaystyle R^2$. If ||$\displaystyle \vec{x}$|| = 2 and ||[$\displaystyle \vec{y}$|| = 3, what, if anything, can we conclude about the possible values of |$\displaystyle \vec{x}^{T}\vec{y}$|?

So this is what I have so far...

$\displaystyle 2 = \sqrt{x_{1}^{2}+x_{2}^{2}}$
$\displaystyle 3 = \sqrt{y_{1}^{2}+y_{2}^{2}}$

After some algebra I got...

$\displaystyle \vec{x}^{T}\vec{y}=(\sqrt{4-x_{2}^{2}}\sqrt{4-x_{1}^{2}})(\sqrt{9-y_{2}^{2}}\sqrt{9-y_{1}^{2}})^{T}$

This is where I'm stuck, what can I determine from this? Thanks in advance!

2. Originally Posted by pakman
Let $\displaystyle \vec{x}$ and $\displaystyle \vec{y}$ be linearly independent vectors in $\displaystyle R^2$. If ||$\displaystyle \vec{x}$|| = 2 and ||[$\displaystyle \vec{y}$|| = 3, what, if anything, can we conclude about the possible values of |$\displaystyle \vec{x}^{T}\vec{y}$|?

So this is what I have so far...

$\displaystyle 2 = \sqrt{x_{1}^{2}+x_{2}^{2}}$
$\displaystyle 3 = \sqrt{y_{1}^{2}+y_{2}^{2}}$

After some algebra I got...

$\displaystyle \vec{x}^{T}\vec{y}=(\sqrt{4-x_{2}^{2}}\sqrt{4-x_{1}^{2}})(\sqrt{9-y_{2}^{2}}\sqrt{9-y_{1}^{2}})^{T}$

This is where I'm stuck, what can I determine from this? Thanks in advance!
If I'm not mistaken this looks like it's an Euclidean dot product, so
$\displaystyle |\vec{x}^T \vec{y}| = ||\vec{x}||~||\vec{y}||~|cos(\theta)|$ which ranges from 0 to 2 * 3 = 6.

-Dan

3. Originally Posted by topsquark
If I'm not mistaken this looks like it's an Euclidean dot product, so
$\displaystyle |\vec{x}^T \vec{y}| = ||\vec{x}||~||\vec{y}||~|cos(\theta)|$ which ranges from 0 to 2 * 3 = 6.

-Dan
How do you figure that it ranges from 0 to 6? I am a bit confused...

4. Originally Posted by pakman
How do you figure that it ranges from 0 to 6? I am a bit confused...
$\displaystyle ||\vec{x}||$ and $\displaystyle ||\vec{y}||$ are constants. And $\displaystyle |cos(\theta)|$ ranges from 0 to 1.

-Dan

5. Originally Posted by topsquark
$\displaystyle ||\vec{x}||$ and $\displaystyle ||\vec{y}||$ are constants. And $\displaystyle |cos(\theta)|$ ranges from 0 to 1.

-Dan
Ahh okay makes sense now thanks!