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Glacius
Linear algebraConcept reference

Euclidean length

Euclidean length is the nonnegative square root of the sum of squared coordinates.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Euclidean length measures a vector’s straight arrow
  3. Signs disappear when coordinates are squared
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Euclidean length measures a vector’s straight arrow#

Euclidean length measures a vector’s straight arrow. For v=(3,4)v=(3,4), perpendicular coordinate changes form a right triangle.

The vector from (0,0) to (3,4) has perpendicular coordinate legs of lengths3 and4. Its straight-line length is5. The x and y scales are equal.The vector from (0,0) to (3,4) has perpendicular coordinate legs of lengths3 and4. Its straight-line length is5. The x and y scales are equal.
Figure 1Both axes use the same unit scale.
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For v=(3,4)v=(3,4), square the two perpendicular contributions: 32=93^2=9 and 42=164^2=16. Their sum gives the squared length.

Take the nonnegative square root.

∥v∥2=9+16=5\|v\|_2=\sqrt{9+16}=5

The subscript 22 names Euclidean length. For any coordinate vector, square every entry, add, then take the square root. The dot product with itself gives that sum.

Using the dot product you already know.

∥v∥2=v⋅v\|v\|_2=\sqrt{v\cdot v}
Check your reasoning

What is the Euclidean length of v=(−6,8)v=(-6,8)?

Show answer and explanation
10

(−6)2+82=100(-6)^2+8^2=100. The nonnegative square root is 1010.

02Signs disappear when coordinates are squared#

Signs disappear when coordinates are squared. For (−3,4)(-3,4), the length is still 55. The zero vector has length 00; every nonzero vector has positive length.

The zero vector has zero length.

∥(0,0)∥2=0\|(0,0)\|_2=0
Check your reasoning

A grid move changes x by 5 and y by −12. What is its straight-line length?

Show answer and explanation
13

The squared changes contribute 25+144=16925+144=169, so the length is 1313.

03Check your understanding#

Check your reasoning

For v=(1,2,2)v=(1,2,2), length 99 omits which step?

  1. ASquare entries.
  2. BAdd squares.
  3. CTake a square root.
Show answer and explanation
Take a square root.

The squared sum is 99; the length is 33.

Key takeaway

Square each coordinate, add, then take the nonnegative square root.

  • Compute the Euclidean length of a vector

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Euclidean length.” Math behind ML. /learn/la-length