Dot Product Math Example 2

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Example 2

medium
Find the angle between a=⟨1,0⟩\mathbf{a} = \langle 1, 0 \rangle and b=⟨1,1⟩\mathbf{b} = \langle 1, 1 \rangle.

Solution

  1. 1
    Step 1: a⋅b=1(1)+0(1)=1\mathbf{a} \cdot \mathbf{b} = 1(1) + 0(1) = 1.
  2. 2
    Step 2: ∥a∥=1\|\mathbf{a}\| = 1, ∥b∥=2\|\mathbf{b}\| = \sqrt{2}.
  3. 3
    Step 3: cos⁡θ=11⋅2=12\cos\theta = \frac{1}{1 \cdot \sqrt{2}} = \frac{1}{\sqrt{2}}.
  4. 4
    Step 4: θ=45°\theta = 45°.

Answer

45°45°
The dot product relates to the angle: cos⁡θ=a⋅b∥a∥∥b∥\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}. This is one of the most important formulas in vector math.

About Dot Product

The dot product of two vectors a=⟨a1,a2⟩\mathbf{a} = \langle a_1, a_2 \rangle and b=⟨b1,b2⟩\mathbf{b} = \langle b_1, b_2 \rangle is the scalar a⋅b=a1b1+a2b2\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. Equivalently, a⋅b=∥a∥∥b∥cos⁡θ\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta, where θ\theta is the angle between the vectors.

Learn more about Dot Product →

More Dot Product Examples

Example 1 easy

Find [formula].

Example 3 easy

Are [formula] and [formula] perpendicular?

Example 4 hard

Find [formula].

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