Compute the 1-norm, 2-norm, and infinity norm for two vectors in R^4, and find the distances between them using these norms.

Vault Of Euclid

Computing the 1, 2, and Infinity Norms and Distances in R^4

Inner product spaces and orthogonality warmup

Consider vectors \( u = (1,3, -6, 4) \) and \( v = (3, -5,1, -2) \) in \( \mathbb{R}^4 \). Find (a) \( \|u\|_\infty \) and \( \|v\|_\infty \), (b) \( \|u\|_1 \) and \( \|v\|_1 \), (c) \( \|u\|_2 \) and \( \|v\|_2 \), (d) \( d_\infty(u, v), d_1(u, v), d_2(u, v) \).

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What you need first

Basic vector arithmetic (addition and subtraction)
Absolute values

What this problem tests

Tests the definitions and basic computation of the standard p-norms and their induced metrics.

This is a standard introductory exercise in linear algebra or functional analysis to familiarize students with different ways of measuring the size of a vector and the distance between points.

Computing the 1, 2, and Infinity Norms and Distances in R^4

What you need first

What this problem tests

More on Inner product spaces and orthogonality