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Norms and Unit Vectors

Straight-line, taxicab or largest component: three ways to say how long a vector is, three different "unit circles", and one way to make any vector length 1.

Step 1 of 27

Speed

How long is v = (1.5, 2)? That sounds like it has one answer, but "length" is a choice. A norm is a rule for measuring vectors, and different rules suit different jobs.

v =

\textcolor{#c084fc}{\|\mathbf{v}\|_1} = |1.5| + |2| = 3.5 \qquad \textcolor{#38bdf8}{\|\mathbf{v}\|_2} = \sqrt{1.5^2 + 2^2} = 2.5 \qquad \textcolor{#fb923c}{\|\mathbf{v}\|_\infty} = \max(|1.5|, |2|) = 2

Purple is L1, blue is L2, orange is L∞. Solid shapes are the unit "circles"; dashed ones pass through v's tip.

Norms and Unit Vectors

Straight-line, taxicab or largest component: three ways to say how long a vector is, three different "unit circles", and one way to make any vector length 1.

Step 1 of 27

Speed

How long is v = (1.5, 2)? That sounds like it has one answer, but "length" is a choice. A norm is a rule for measuring vectors, and different rules suit different jobs.

v =

\textcolor{#c084fc}{\|\mathbf{v}\|_1} = |1.5| + |2| = 3.5 \qquad \textcolor{#38bdf8}{\|\mathbf{v}\|_2} = \sqrt{1.5^2 + 2^2} = 2.5 \qquad \textcolor{#fb923c}{\|\mathbf{v}\|_\infty} = \max(|1.5|, |2|) = 2

Purple is L1, blue is L2, orange is L∞. Solid shapes are the unit "circles"; dashed ones pass through v's tip.

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Norms and Unit Vectors

See also

Norms and Unit Vectors

See also