Inner Product Spaces

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

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

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Originally Posted by pakman

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

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

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

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.