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Glacius
Linear algebraConcept reference

Vector differences

The displacement from an initial coordinate vector to a final coordinate vector is final minus initial.

1 min read2 figuresSources & further reading
On this page 8 sections
  1. Overview
  2. Displacement is the coordinate change from a stated start to a stated end
  3. From B=(4,1) to A=(1,2), displacement is (-3,1)
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Displacement is the coordinate change from a stated start to a stated end#

Displacement is the coordinate change from a stated start to a stated end. From A=(1,2)A=(1,2) to B=(4,1)B=(4,1), the arrow points three units right and one down.

On equal-scale x/y axes, the arrow starts at A=(1,2) and ends at B=(4,1). It changes x by +3 and y by -1.On equal-scale x/y axes, the arrow starts at A=(1,2) and ends at B=(4,1). It changes x by +3 and y by -1.
Figure 1On equal-scale x/y axes, the arrow starts at A=(1,2) and ends at B=(4,1). It changes x by +3 and y by -1.
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For start A=(1,2)A=(1,2) and end B=(4,1)B=(4,1), compute B+(−1)AB+(-1)A. The opposite of A is (−1,−2)(-1,-2). Add it to B to get (3,−1)(3,-1). This is written B−AB-A, or end minus start.

Check your reasoning

From A=(2,1)A=(2,1) to B=(−1,3)B=(-1,3). Find displacement.

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

End minus start gives (−3,2)(-3,2).

02From B=(4,1) to A=(1,2), displacement is (-3,1)#

From B=(4,1)B=(4,1) to A=(1,2)A=(1,2), displacement is (−3,1)(-3,1). Reversing endpoints reverses every sign. Matching endpoints give zero change.

The same equal-scale coordinates A=(1,2), B=(4,1), with the arrow reversed from B to A. Its coordinate change is (-3,1).The same equal-scale coordinates A=(1,2), B=(4,1), with the arrow reversed from B to A. Its coordinate change is (-3,1).
Figure 2The same equal-scale coordinates A=(1,2), B=(4,1), with the arrow reversed from B to A. Its coordinate change is (-3,1).
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Check your reasoning

From A=(1,−2)A=(1,-2) to B=(1,4)B=(1,4). Fill displacement (0,□)(0,\square).

Show answer and explanation
6

The vertical change is 4+2=64+2=6.

03Check your understanding#

Check your reasoning

Counts (red, green, blue). Before: (2,4,1)(2,4,1). After: (5,2,1)(5,2,1). Change vector?

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

After minus before gives (3,−2,0)(3,-2,0).

Key takeaway

Compute end minus start; reversing the trip reverses every sign.

  • Compute an ordered displacement between two vectors

Further questions

Is displacement the length of the journey?
Displacement records signed coordinate change from start to end. It does not record the route taken. Its length is a separate quantity.

Sources & further reading

  1. [1]
  2. [2]

Reference this concept

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

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