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Glacius
Linear algebraConcept reference

Vector coordinates

A coordinate vector is an ordered list of numbers interpreted relative to a specified coordinate system.

1 min read2 figuresSources & further reading
On this page 8 sections
  1. Overview
  2. A vector records numbers in a fixed order
  3. Coordinate names determine the order
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01A vector records numbers in a fixed order#

A vector records numbers in a fixed order. With coordinates (cats, dogs), (3,1)(3,1) records 3 cats and 1 dog. Each number is an entry of the vector.

In coordinate order cats then dogs, the vector (3,1) records 3 cats and 1 dog.In coordinate order cats then dogs, the vector (3,1) records 3 cats and 1 dog.
Figure 1In coordinate order cats then dogs, the vector (3,1) records 3 cats and 1 dog.
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For spatial coordinates (x, y), v=(3,−2)v=(3,-2) points from the origin three units right and two down. Each sign belongs to its named axis.

Equal-scale x and y axes meet at the origin. The vector (3,-2) ends at x=3,y=-2, three units right and two down.Equal-scale x and y axes meet at the origin. The vector (3,-2) ends at x=3,y=-2, three units right and two down.
Figure 2Equal-scale x and y axes meet at the origin. The vector (3,-2) ends at x=3,y=-2, three units right and two down.
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Vectors can record more than two quantities. In order (tokens, links, errors), (8,2,0)(8,2,0) records 8 tokens, 2 links and 0 errors. It has three entries, so its dimension is three. A zero still occupies a slot.

Check your reasoning

Order: (tokens, links, errors). Vector: (6,4,0)(6,4,0). Value for links?

Show answer and explanation
4

Links occupy the second slot, which contains 44.

02Coordinate names determine the order#

Coordinate names determine the order. With 3 cats and 1 dog, order (cats, dogs) gives (3,1)(3,1). Order (dogs, cats) gives (1,3)(1,3). The recorded quantities stay the same when their labels move with them.

Check your reasoning

Order: (red, green, blue). Values: blue=5, red=2, green=0. Which vector?

  1. A(5,0,2)(5,0,2)
  2. B(2,0,5)(2,0,5)
  3. C(2,5)(2,5)
Show answer and explanation
(2,0,5)(2,0,5)

Red first, green second, blue third: (2,0,5)(2,0,5).

03Check your understanding#

Check your reasoning

Order: (width, height). Values: height=7, width=4. Which vector?

  1. A(7,4)(7,4)
  2. B(4,4)(4,4)
  3. C(4,7)(4,7)
Show answer and explanation
(4,7)(4,7)

Width takes the first slot: (4,7)(4,7).

Key takeaway

Keep every value with its coordinate label, including zero entries.

  • Interpret a vector using its declared coordinate order

Further questions

Can a vector have more than three entries?
Yes. Its entries can record any finite number of named quantities. A drawing in physical space is only one use of vectors.

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

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

Glacius. “Vector coordinates.” Math behind ML. /learn/la-coordinates