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Glacius
Linear algebraConcept reference

Vector addition

The sum of two equal-dimension vectors is the vector of their coordinate-wise sums.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. Vector addition combines matching coordinates
  3. Both vectors need the same number of entries
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Vector addition combines matching coordinates#

Vector addition combines matching coordinates. For order (x, y), adding (3,1)(3,1) and (1,2)(1,2) combines the two x-changes and the two y-changes.

For (3,1)+(1,2)(3,1)+(1,2), the x-entry is 3+1=43+1=4 and the y-entry is 1+2=31+2=3. Following the two arrows reaches the same endpoint as the sum (4,3)(4,3).

With equal x/y scale, u=(3,1) runs from (0,0) to (3,1); v=(1,2) continues to (4,3). The sum arrow connects (0,0) directly to (4,3).With equal x/y scale, u=(3,1) runs from (0,0) to (3,1); v=(1,2) continues to (4,3). The sum arrow connects (0,0) directly to (4,3).
Figure 1With equal x/y scale, u=(3,1) runs from (0,0) to (3,1); v=(1,2) continues to (4,3). The sum arrow connects (0,0) directly to (4,3).
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Check your reasoning

Add (1,3)(1,3) and (2,−1)(2,-1).

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

Add x to x and y to y: (3,2)(3,2).

02Both vectors need the same number of entries#

Both vectors need the same number of entries. A two-entry vector has no third coordinate to pair with a three-entry vector. For named quantities, also use matching labels and units in the same order.

Check your reasoning

Add (4,−2)(4,-2) and (−1,5)(-1,5). Fill (3,□)(3,\square).

Show answer and explanation
3

The second entry is −2+5=3-2+5=3.

03Check your understanding#

Check your reasoning

Counts (books, pens): (2,4)(2,4) and (3,1)(3,1). Combined vector?

  1. A(5,4)(5,4)
  2. B(5,5)(5,5)
  3. C(6,4)(6,4)
Show answer and explanation
(5,5)(5,5)

Combine books and pens separately: (5,5)(5,5).

Key takeaway

Add matching coordinates and preserve their order.

  • Add full vectors with matching coordinate meanings

Further questions

What if the coordinate orders differ?
Rewrite the data in a common labeled order before adding. Same dimension alone does not make different labels or units compatible.

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

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

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