Angle between Vectors Calculator
The calculator will find the angle (in radians and degrees) between the two vectors and will show the work.
Your Input
Calculate the angle between the vectors $$$\mathbf{\vec{u}} = \left\langle 5, -2, 3\right\rangle$$$ and $$$\mathbf{\vec{v}} = \left\langle -4, 5, 7\right\rangle$$$.
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:
$$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{38}$$$ (for steps, see vector length calculator).
$$$\mathbf{\left\lvert\vec{v}\right\rvert} = 3 \sqrt{10}$$$ (for steps, see vector length calculator).
Finally, the angle is given by $$$\cos{\left(\phi \right)} = \frac{\mathbf{\vec{u}}\cdot \mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert} \mathbf{\left\lvert\vec{v}\right\rvert}} = \frac{-9}{\left(\sqrt{38}\right)\cdot \left(3 \sqrt{10}\right)} = - \frac{3 \sqrt{95}}{190}$$$ (in case of complex numbers, we need to take the real part of the dot product).
$$$\phi = \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)} = \left(\frac{180 \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)}}{\pi}\right)^0$$$