L2 Norm

Definition

The L2 norm measures the length of a vector.

For:

v = [3, 4]

the L2 norm is:

5

L2 norm = the usual geometric length of a vector.

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Formula

For a two-component vector:

v = [a, b]

the L2 norm is:

sqrt(a^2 + b^2)

Example:

v = [3, 4]

sqrt(3^2 + 4^2)
= sqrt(9 + 16)
= sqrt(25)
= 5

The L2 norm combines component sizes into one length.

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More Components

For three components:

v = [a, b, c]

the L2 norm is:

sqrt(a^2 + b^2 + c^2)

Example:

v = [2, 3, 6]

sqrt(2^2 + 3^2 + 6^2)
= sqrt(4 + 9 + 36)
= sqrt(49)
= 7

The same pattern extends to more components.

L2 norm works by squaring each component, adding, then taking the square root.

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L2 Norm vs Euclidean Distance

L2 norm measures the length of one vector from the origin.

Euclidean distance measures separation between two points.

Example:

norm([3, 4]) = 5

Distance:

A = [1, 2]
B = [4, 6]
B - A = [3, 4]
distance(A, B) = norm([3, 4]) = 5

Euclidean distance is the L2 norm of a difference vector.

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L2 Norm Is Non-Negative

The L2 norm is never negative.

Examples:

norm([3, 4]) = 5
norm([-3, -4]) = 5
norm([0, 0]) = 0

Negative components still contribute positive size because they are squared.

L2 norm measures size, not sign.

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Zero Norm

Only the zero vector has L2 norm 0.

Examples:

norm([0, 0]) = 0
norm([0, 0, 0]) = 0

If any component is nonzero, the norm is greater than 0.

Zero length means every component is zero.

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Why L2 Norm Matters

The L2 norm gives a standard way to measure vector length.

It helps answer:

How large is this vector?
How far is this point from the origin?
How much difference is there between two points?

L2 norm is the basic length measure used by Euclidean geometry.

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Final Summary

• The L2 norm measures vector length.
• It uses square, sum, and square root.
• It works for vectors with any number of components.
• It is always non-negative.
• Only the zero vector has L2 norm 0.
• Euclidean distance is the L2 norm of a difference vector.