# Angle between Vectors Calculator

The calculator will find the angle (in radians and degrees) between the two vectors and will show the work.

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If you have two-dimensional vectors, set the third coordinates equal to $0$ or leave them empty.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Calculate the angle $\phi$ between the vectors $\mathbf{\vec{u}} = \left(5, -2, 3\right)$ and $\mathbf{\vec{v}} = \left(-4, 5, 7\right)$.

## Solution

First, calculate the dot product: $\mathbf{\vec{u}}\cdot \mathbf{\vec{v}} = -9$ (for steps, see dot product calculator).

Next, find the lengths of the vectors:

$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}} = \sqrt{5^{2} + \left(-2\right)^{2} + 3^{2}} = \sqrt{38}$ (for steps, see vector length calculator).

$\mathbf{\left\lvert\vec{v}\right\rvert} = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}} = \sqrt{\left(-4\right)^{2} + 5^{2} + 7^{2}} = 3 \sqrt{10}$ (for steps, see vector length calculator).

Finally, the angle is given by $\cos{\left(\phi \right)} = \frac{\mathbf{\vec{u}}\cdot \mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert} \mathbf{\left\lvert\vec{v}\right\rvert}} = \frac{-9}{\left(\sqrt{38}\right)\cdot \left(3 \sqrt{10}\right)} = - \frac{3 \sqrt{95}}{190}.$

$\phi = \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)} = \left(\frac{180 \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)}}{\pi}\right)^0$

$\phi = \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)}\approx 1.725307134097968$A
$\phi = \left(\frac{180 \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)}}{\pi}\right)^0\approx 98.852817147625106^0$A