Vectors and their components

About this lesson

Precalculus › Vectors

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Vectors and their components

Discover

You walk from P=(1,2)P = (1, 2) to Q=(4,6)Q = (4, 6). How far do you go to the right, and how far up?

  1. 3 to the right and 4 up
  2. 3 to the left and 4 down

Key idea

A vector has a length and a direction, but no fixed place. We write it by its components.

From P=(x1,y1)P = (x_1, y_1) to Q=(x2,y2)Q = (x_2, y_2): PQ→=⟨x2−x1, y2−y1⟩\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle (end minus start). Two vectors with the same components are equal, wherever they start.

Worked example

Find PQ→\overrightarrow{PQ} for P=(5,−1)P = (5, -1) and Q=(2,3)Q = (2, 3).

2−5=−32 - 5 = -3
End minus start, horizontally.
3−(−1)=43 - (-1) = 4
End minus start, vertically.
PQ→=⟨−3,4⟩\overrightarrow{PQ} = \langle -3, 4 \rangle
Write the components.

Your turn, guided

Your turn: P=(2,7)P = (2, 7) and Q=(6,1)Q = (6, 1).

Horizontal: 44
6 − 2.
Vertical: −6-6, so PQ→=⟨4,−6⟩\overrightarrow{PQ} = \langle 4, -6 \rangle
1 − 7.

Practice ★

  1. P=(2,4)P = (2, 4) and Q=(4,−6)Q = (4, -6). Find the components of PQ→\overrightarrow{PQ}.
  2. P=(5,1)P = (5, 1) and Q=(7,−2)Q = (7, -2). Find the components of PQ→\overrightarrow{PQ}.
  3. P=(2,3)P = (2, 3) and Q=(9,−8)Q = (9, -8). Find the components of PQ→\overrightarrow{PQ}.

Watch out!

Practice ★★

  1. The vector ⟨2,1⟩\langle 2, 1 \rangle starts at the point (1,1)(1, 1). Tap the point where it ends.
  2. The vector ⟨2,3⟩\langle 2, 3 \rangle starts at the point (1,−3)(1, -3). Tap the point where it ends.
  3. The vector ⟨3,1⟩\langle 3, 1 \rangle starts at the point (−2,−1)(-2, -1). Tap the point where it ends.
  1. P=(3,3)P = (3, 3) and Q=(8,8)Q = (8, 8). Which vector is PQ→\overrightarrow{PQ}?
    1. ⟨5,5⟩\langle 5, 5 \rangle
    2. ⟨−5,−5⟩\langle -5, -5 \rangle
    3. ⟨11,11⟩\langle 11, 11 \rangle
  2. P=(2,2)P = (2, 2) and Q=(8,6)Q = (8, 6). Which vector is PQ→\overrightarrow{PQ}?
    1. ⟨6,4⟩\langle 6, 4 \rangle
    2. ⟨−6,−4⟩\langle -6, -4 \rangle
    3. ⟨10,8⟩\langle 10, 8 \rangle

Exit ticket

Three questions to check you have got it.

  1. P=(2,4)P = (2, 4) and Q=(4,−6)Q = (4, -6). Find the components of PQ→\overrightarrow{PQ}.
  2. P=(5,1)P = (5, 1) and Q=(7,−2)Q = (7, -2). Find the components of PQ→\overrightarrow{PQ}.
  3. P=(2,3)P = (2, 3) and Q=(9,−8)Q = (9, -8). Find the components of PQ→\overrightarrow{PQ}.

Summary card

Vectors and components

  • PQ→=⟨x2−x1, y2−y1⟩\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle: end minus start.
  • Same components, same vector, wherever it starts.

Watch out: start minus end gives the opposite vector.

End of the lesson

Next lesson: Magnitude and direction of a vector Back to the chapter

Answers: Vectors and their components

A

  1. −10-10
  2. −3-3
  3. −11-11

B

  1. (3,2)(3, 2)
  2. (3,0)(3, 0)
  3. (1,0)(1, 0)

C

  1. ⟨5,5⟩\langle 5, 5 \rangle
  2. ⟨6,4⟩\langle 6, 4 \rangle

D

  1. −10-10
  2. −3-3
  3. −11-11

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