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Determine the angle between two vectors

The angle between two nonzero vectors is the smallest angle from one vector’s direction to the other, determined from their dot product by cos⁡θ=a⋅b∥a∥∥b∥\cos\theta=\frac{\mathbf a\cdot\mathbf b}{\|\mathbf a\|\|\mathbf b\|}, using component form in two or three dimensions and 0≤θ≤180∘0\leq\theta\leq180^\circ (or 0≤θ≤π0\leq\theta\leq\pi). The signs of the dot product identify acute, right, or obtuse relationships; the zero vector has no direction and therefore no defined angle, supporting geometric reasoning about perpendicularity, projections, and vector applications.

Detailed Explanation: Determine the angle between two vectors

For two nonzero vectors a\mathbf a and b\mathbf b, the angle θ\theta between them is found using the dot product formula:

cos⁡θ=a⋅b ∣a ∣ ∣b ∣.\cos\theta=\frac{\mathbf a\cdot\mathbf b}{\ \vert \mathbf a\ \vert \ \vert \mathbf b\ \vert }.

The angle is always chosen between 0∘0^\circ and 180∘180^\circ.

Worked example

Find the angle between

a=(21)andb=(13).\mathbf a=\begin{pmatrix}2\\1\end{pmatrix} \quad\text{and}\quad \mathbf b=\begin{pmatrix}1\\3\end{pmatrix}.

Step 1: Find the dot product.

Multiply corresponding components and add:

a⋅b=(2)(1)+(1)(3)=2+3=5.\mathbf a\cdot\mathbf b=(2)(1)+(1)(3)=2+3=5.

Step 2: Find the length of each vector.

For a\mathbf a:

 ∣a ∣=22+12=5.\ \vert \mathbf a\ \vert =\sqrt{2^2+1^2}=\sqrt5.

For b\mathbf b:

 ∣b ∣=12+32=10.\ \vert \mathbf b\ \vert =\sqrt{1^2+3^2}=\sqrt{10}.

Step 3: Substitute into the formula.

cos⁡θ=5(5)(10)=550=12.\cos\theta =\frac{5}{(\sqrt5)(\sqrt{10})} =\frac{5}{\sqrt{50}} =\frac{1}{\sqrt2}.

Step 4: Use the inverse cosine.

θ=cos⁡−1(12)=45∘.\theta=\cos^{-1}\left(\frac{1}{\sqrt2}\right)=45^\circ.

Therefore, the angle between the vectors is

45∘.\boxed{45^\circ}.

Because the dot product was positive, the angle is acute. In general:

  • If a⋅b>0\mathbf a\cdot\mathbf b>0, the angle is acute.
  • If a⋅b=0\mathbf a\cdot\mathbf b=0, the vectors are perpendicular, so the angle is 90∘90^\circ.
  • If a⋅b<0\mathbf a\cdot\mathbf b<0, the angle is obtuse.

The zero vector cannot be used because it has no direction, so its angle with another vector is undefined.

Learn by doing: Determine the angle between two vectors

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