I have this for my homework and i need help with this question: For a 2D vector, is norm2 > norm1 possible? Characterize all 2D vectors for which the norm2 is equal to the norm1. Explain graphically.
2026-05-05 12:01:35.1777982495
Equivalence of a Vector Norm
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For $(x,y) \in \mathbb R^2$ we have
$$||(x,y)||_1 =|x|+|y| \ge \sqrt{x^2+y^2}=||(x,y)||_2,$$
since
$(|x|+|y|)^2=x^2+2|x||y|+y^2 \ge x^2+y^2$.
We have the equality
$$||(x,y)||_1 =|x|+|y| = \sqrt{x^2+y^2}=||(x,y)||_2$$
$ \iff$
$(|x|+|y|)^2=x^2+2|x||y|+y^2 = x^2+y^2 \iff xy=0$.