Calculate the angle between two vectors

This page calculates the angle between two vectors. To perform the calculation enter the X/Y coordinates of the two vectors. Then click on the 'Calculate' button.

Vector angle calculator 2 Elements 3 Elements  Input Delete Entries Vector a Vector b x y Decimal places 0 1 2 3 4 6 8 10  Results Angle

Formulas of angle between vectors

\(\displaystyle cos(θ) = \frac{\vec{a}·\vec{b}}{|\vec{a}|·|\vec{b}|}\)

The scalar product of the two vectors is in the numerator and the product of the absolute value (lengths) of the vectors is in the denominator.

Example

Calculation of the angle between \(\displaystyle \vec{a} = \left[\matrix{4 \\ 5 }\right]\)   and   \(\displaystyle \vec{b} = \left[\matrix{-7 \\ \;\;2 }\right]\)

1. Calculate scalar product

\(\displaystyle \vec{a}·\vec{b}=4·(-7)+5·2=(-28)+10=-18\)

2. Calculate lengths of vectors

\(\displaystyle |\vec{a}| = \sqrt{4^2+5^2} =\sqrt{16+25}=\sqrt{41}\)
\(\displaystyle |\vec{b}| = \sqrt{-7^2+2^2} =\sqrt{49+4}=\sqrt{53}\)

3. calculate formula

\(\displaystyle cos(θ)=\frac{-18}{\sqrt{41}\cdot\sqrt{53}} ≈ 0.3861\)

\(\displaystyle θ=cos^{-1}(0.3861)≈112.71°\)