# Inner Product Spaces

• May 29th 2008, 01:51 PM
pakman
Inner Product Spaces
Let $x=(5,2,4)^T$ and $y=(3,3,2)^T$. Compute $||x-y||_1$, $||x-y||_2$, and $||x-y||_{\infty}$. Under which norm are the two vectors closest together? Under which norm are they farthest apart?

I computed $||x-y||_1=5$ and $||x-y||_2=3$. The infinity is throwing me off however. And for the distance would I just compare the square roots of the vectors?
• May 30th 2008, 12:09 AM
Opalg
Quote:

Originally Posted by pakman
Let $x=(5,2,4)^T$ and $y=(3,3,2)^T$. Compute $||x-y||_1$, $||x-y||_2$, and $||x-y||_{\infty}$. Under which norm are the two vectors closest together? Under which norm are they farthest apart?

I computed $||x-y||_1=5$ and $||x-y||_2=3$. The infinity is throwing me off however. And for the distance would I just compare the square roots of the vectors?

If $x=(x_1,x_2,x_3)^{\textsc{t}}$ and $y=(y_1,y_2,y_3)^{\textsc{t}}$ then $||x-y||_{\infty} = \max\{|x_1-y_1|,|x_2-y_2|,|x_3-y_3|\}$.

The last parts of the question simply seem to be asking you to say which of the three numbers that you computed in the first part of the question is the smallest, and which is the greatest.