Add, subtract and scale vectors

About this lesson

Precalculus › Vectors

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Add, subtract and scale vectors

Warm-up

Three quick questions from earlier lessons.

Discover

You walk ⟨3,1⟩\langle 3, 1 \rangle, then ⟨1,2⟩\langle 1, 2 \rangle. Where are you, compared with your start?

  1. ⟨4,3⟩\langle 4, 3 \rangle
  2. ⟨2,−1⟩\langle 2, -1 \rangle

Key idea

Vectors combine component by component.

⟨a,b⟩+⟨c,d⟩=⟨a+c, b+d⟩\langle a, b \rangle + \langle c, d \rangle = \langle a + c,\ b + d \rangle;  ⟨a,b⟩−⟨c,d⟩=⟨a−c, b−d⟩\ \langle a, b \rangle - \langle c, d \rangle = \langle a - c,\ b - d \rangle;  k⟨a,b⟩=⟨ka, kb⟩\ k\langle a, b \rangle = \langle ka,\ kb \rangle. Geometrically, u⃗+v⃗\vec{u} + \vec{v} goes tip to tail. In general ∣u⃗+v⃗∣≠∣u⃗∣+∣v⃗∣|\vec{u} + \vec{v}| \ne |\vec{u}| + |\vec{v}|.

Worked example

u⃗=⟨2,−1⟩\vec{u} = \langle 2, -1 \rangle and v⃗=⟨−3,4⟩\vec{v} = \langle -3, 4 \rangle. Find 3u⃗+v⃗3\vec{u} + \vec{v}.

3u⃗=⟨6,−3⟩3\vec{u} = \langle 6, -3 \rangle
Scale u.
3u⃗+v⃗=⟨6−3, −3+4⟩=⟨3,1⟩3\vec{u} + \vec{v} = \langle 6 - 3,\ -3 + 4 \rangle = \langle 3, 1 \rangle
Add component by component.

Your turn, guided

Your turn: u⃗=⟨4,1⟩\vec{u} = \langle 4, 1 \rangle, v⃗=⟨1,−2⟩\vec{v} = \langle 1, -2 \rangle. Find u⃗−2v⃗\vec{u} - 2\vec{v}.

Horizontal: 4−2=24 - 2 = 2
2v = ⟨2, −4⟩.
Vertical: 55, so u⃗−2v⃗=⟨2,5⟩\vec{u} - 2\vec{v} = \langle 2, 5 \rangle
1 − (−4).

Practice ★

  1. u⃗=⟨4,6⟩\vec{u} = \langle 4, 6 \rangle and v⃗=⟨1,−4⟩\vec{v} = \langle 1, -4 \rangle. Find 4u⃗−v⃗ 4\vec{u} - \vec{v}.
  2. u⃗=⟨3,1⟩\vec{u} = \langle 3, 1 \rangle and v⃗=⟨4,−6⟩\vec{v} = \langle 4, -6 \rangle. Find 4u⃗−v⃗ 4\vec{u} - \vec{v}.
  3. u⃗=⟨5,2⟩\vec{u} = \langle 5, 2 \rangle and v⃗=⟨3,−5⟩\vec{v} = \langle 3, -5 \rangle. Find 2u⃗−v⃗ 2\vec{u} - \vec{v}.

Watch out!

Practice ★★

  1. u⃗=⟨4,1⟩\vec{u} = \langle 4, 1 \rangle and v⃗=⟨5,6⟩\vec{v} = \langle 5, 6 \rangle. Find the exact value of ∣u⃗+v⃗∣|\vec{u} + \vec{v}|.
  2. u⃗=⟨5,5⟩\vec{u} = \langle 5, 5 \rangle and v⃗=⟨5,3⟩\vec{v} = \langle 5, 3 \rangle. Find the exact value of ∣u⃗+v⃗∣|\vec{u} + \vec{v}|.
  3. u⃗=⟨1,4⟩\vec{u} = \langle 1, 4 \rangle and v⃗=⟨2,2⟩\vec{v} = \langle 2, 2 \rangle. Find the exact value of ∣u⃗+v⃗∣|\vec{u} + \vec{v}|.
  1. u⃗=⟨2,3⟩\vec{u} = \langle 2, 3 \rangle and v⃗=⟨−3,1⟩\vec{v} = \langle -3, 1 \rangle, both starting at the origin. Tap the end point of u⃗+v⃗\vec{u} + \vec{v}.
  2. u⃗=⟨4,1⟩\vec{u} = \langle 4, 1 \rangle and v⃗=⟨−2,2⟩\vec{v} = \langle -2, 2 \rangle, both starting at the origin. Tap the end point of u⃗+v⃗\vec{u} + \vec{v}.

Exit ticket

Three questions to check you have got it.

  1. u⃗=⟨4,1⟩\vec{u} = \langle 4, 1 \rangle and v⃗=⟨5,6⟩\vec{v} = \langle 5, 6 \rangle. Find the exact value of ∣u⃗+v⃗∣|\vec{u} + \vec{v}|.
  2. u⃗=⟨5,5⟩\vec{u} = \langle 5, 5 \rangle and v⃗=⟨5,3⟩\vec{v} = \langle 5, 3 \rangle. Find the exact value of ∣u⃗+v⃗∣|\vec{u} + \vec{v}|.
  3. u⃗=⟨1,4⟩\vec{u} = \langle 1, 4 \rangle and v⃗=⟨2,2⟩\vec{v} = \langle 2, 2 \rangle. Find the exact value of ∣u⃗+v⃗∣|\vec{u} + \vec{v}|.

Summary card

Vector operations

  • Add, subtract, scale: component by component.
  • u⃗+v⃗\vec{u} + \vec{v}: tip to tail.
  • ∣u⃗+v⃗∣|\vec{u} + \vec{v}|: add first, then take the length.

Watch out: ∣u⃗+v⃗∣≠∣u⃗∣+∣v⃗∣|\vec{u} + \vec{v}| \ne |\vec{u}| + |\vec{v}| unless they point the same way.

End of the lesson

Next lesson: Components from magnitude and direction; unit vectors Back to the chapter

Answers: Add, subtract and scale vectors

A

  1. 2828
  2. 1010
  3. 99

B

  1. 130\sqrt{130}
  2. 2412 \sqrt{41}
  3. 353 \sqrt{5}

C

  1. (−1,4)(-1, 4)
  2. (2,3)(2, 3)

D

  1. 130\sqrt{130}
  2. 2412 \sqrt{41}
  3. 353 \sqrt{5}

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