# 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

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