The dot product and the angle between vectors

About this lesson

Precalculus › Vectors

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The dot product and the angle between vectors

Warm-up

Three quick questions from earlier lessons.

Discover

⟨2,1⟩\langle 2, 1 \rangle and ⟨−1,2⟩\langle -1, 2 \rangle look perpendicular. Multiply matching components and add: 2⋅(−1)+1⋅22 \cdot (-1) + 1 \cdot 2. What do you get?

  1. 00
  2. ⟨−2,2⟩\langle -2, 2 \rangle

Key idea

The dot product of two vectors is a number.

⟨a,b⟩⋅⟨c,d⟩=ac+bd\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd. Also u⃗⋅v⃗=∣u⃗∣ ∣v⃗∣cos⁡θ\vec{u} \cdot \vec{v} = |\vec{u}|\,|\vec{v}|\cos\theta, where θ\theta is the angle between them, so cos⁡θ=u⃗⋅v⃗∣u⃗∣ ∣v⃗∣\cos\theta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}|\,|\vec{v}|}. Two non-zero vectors are perpendicular exactly when u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0.

Worked example

Find the angle between u⃗=⟨1,2⟩\vec{u} = \langle 1, 2 \rangle and v⃗=⟨3,1⟩\vec{v} = \langle 3, 1 \rangle.

u⃗⋅v⃗=3+2=5\vec{u} \cdot \vec{v} = 3 + 2 = 5
Multiply and add.
∣u⃗∣ ∣v⃗∣=5 10=52|\vec{u}|\,|\vec{v}| = \sqrt{5}\,\sqrt{10} = 5\sqrt{2}
√5 and √10.
θ=45∘\theta = 45^\circ
cos θ = 5 / (5√2) = √2/2.

Your turn, guided

Your turn: ⟨4,−1⟩⋅⟨2,3⟩\langle 4, -1 \rangle \cdot \langle 2, 3 \rangle.

First products: 88
4 · 2.
⟨4,−1⟩⋅⟨2,3⟩=5\langle 4, -1 \rangle \cdot \langle 2, 3 \rangle = 5
8 + (−1)(3).

Practice ★

  1. Find ⟨5,2⟩⋅⟨8,−9⟩\langle 5, 2 \rangle \cdot \langle 8, -9 \rangle.
  2. Find ⟨9,8⟩⋅⟨8,−2⟩\langle 9, 8 \rangle \cdot \langle 8, -2 \rangle.
  3. Find ⟨3,4⟩⋅⟨7,−7⟩\langle 3, 4 \rangle \cdot \langle 7, -7 \rangle.
  4. Find ⟨7,5⟩⋅⟨5,−1⟩\langle 7, 5 \rangle \cdot \langle 5, -1 \rangle.

Watch out!

Practice ★★

  1. Are ⟨6,4⟩\langle 6, 4 \rangle and ⟨−8,12⟩\langle -8, 12 \rangle perpendicular?
    1. Yes: their dot product is 0
    2. No: neither of them is the zero vector
    3. No: their components do not add up to 0
  2. Are ⟨2,1⟩\langle 2, 1 \rangle and ⟨−2,4⟩\langle -2, 4 \rangle perpendicular?
    1. Yes: their dot product is 0
    2. No: neither of them is the zero vector
    3. No: their components do not add up to 0
  1. Find the angle between ⟨4,4⟩\langle 4, 4 \rangle and ⟨4,−3⟩\langle 4, -3 \rangle, in degrees, to 1 decimal place.
  2. Find the angle between ⟨3,4⟩\langle 3, 4 \rangle and ⟨5,−4⟩\langle 5, -4 \rangle, in degrees, to 1 decimal place.

Exit ticket

Three questions to check you have got it.

  1. Find ⟨5,2⟩⋅⟨8,−9⟩\langle 5, 2 \rangle \cdot \langle 8, -9 \rangle.
  2. Find ⟨9,8⟩⋅⟨8,−2⟩\langle 9, 8 \rangle \cdot \langle 8, -2 \rangle.
  3. Find ⟨3,4⟩⋅⟨7,−7⟩\langle 3, 4 \rangle \cdot \langle 7, -7 \rangle.

Summary card

The dot product

  • ⟨a,b⟩⋅⟨c,d⟩=ac+bd\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd, a number.
  • cos⁡θ=u⃗⋅v⃗∣u⃗∣ ∣v⃗∣\cos\theta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}|\,|\vec{v}|}.
  • u⃗⋅v⃗=0\vec{u} \cdot \vec{v} = 0: perpendicular.

Watch out: multiply matching components, not crossed ones.

End of the lesson

Next lesson: Vectors in context: velocity and force Back to the chapter

Answers: The dot product and the angle between vectors

A

  1. 2222
  2. 5656
  3. −7-7
  4. 3030

B

  1. Yes: their dot product is 0
  2. Yes: their dot product is 0

C

  1. 180cos⁡−1(210)π\frac{180 \cos^{-1}{\left(\frac{\sqrt{2}}{10} \right)}}{\pi}
  2. 180cos⁡−1(−41205)π\frac{180 \cos^{-1}{\left(- \frac{\sqrt{41}}{205} \right)}}{\pi}

D

  1. 2222
  2. 5656
  3. −7-7

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