Angle Between Two Vectors Calculator

A free online calculator, showing all steps, to calculate the angle $\alpha$ between two vectors in 2D or 3D is presented.

Formulas Used in the Calculator

Let vectors $\vec {U}$ and $\vec {V}$ be defined by their components as follows:
$\vec {U} \; = \; \lt u_{x} , u_{y} , u_{z} \gt$
$\vec {V} \; = \; \lt v_{x} , v_{y} , v_{z} \gt$
The dot product of vectors $\vec {U}$ and $\vec {V}$ is defined as:
$\vec {U} \cdot \vec {V} = | \vec {U} | \cdot | \vec {U} | \cos \alpha$
where $| \vec {U} |$ and $| \vec {U} |$ are the magnitude of vectors $\vec {U}$ and $\vec {V}$ respectively and $\alpha$ is the angle between the two vectors.
and it can be proved that the dot product of the two vectors $\vec {U}$ and $\vec {V}$ is given by a formula involving the components of the two vectors as follows:
$\vec {U} \cdot \vec {V} = u_x \cdot v_x + u_y \cdot v_y + u_z \cdot v_z$
hence the formula for $\cos \alpha$ is given by $$\large \color{red} {\cos \alpha = \dfrac{u_x \cdot v_x + u_y \cdot v_y + u_z \cdot v_z }{| \vec {U} | \cdot | \vec {U} | } }$$ The magnitudes $| \vec {U} |$ and $| \vec {U} |$ are given by
$| \vec {U} | = \sqrt {u_x^2 + u_y^2 + u_z^2 }$
$| \vec {V} | = \sqrt {v_x^2 + v_y^2 + v_z^2 }$
Use the inverse cosine function to express angle $\alpha$ made by the two vectors as $$\large \color{red} {\alpha = \arccos \left (\dfrac{u_x \cdot v_x + u_y \cdot v_y + u_z \cdot v_z }{\sqrt {u_x^2 + u_y^2 + u_z^2 } \cdot \sqrt {v_x^2 + v_y^2 + v_z^2 } } \right) }$$

Use of the Calculator

Enter the components of vectors $\vec {U}$ and $\vec {V}$ and press "Calculate". The outputs are the magnitudes $| \vec {U} |$ and $| \vec {U} |$, the dot product $\vec {U} \cdot \vec {V}$ and angle $\alpha$. You may also enter the number of decimal places required.
Note that you may also use the calculator for 2 D vectors by setting the $z$ components of both vectors equal to zero.

Outputs