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 $$$\phi$$$ 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 $$$\cos{\left(\phi \right)} = \frac{\mathbf{\vec{u}}\cdot \mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert} \mathbf{\left\lvert\vec{v}\right\rvert}} = \frac{-9}{\left(\sqrt{38}\right)\cdot \left(3 \sqrt{10}\right)} = - \frac{3 \sqrt{95}}{190}.$$$
$$$\phi = \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)} = \left(\frac{180 \operatorname{acos}{\left(- \frac{3 \sqrt{95}}{190} \right)}}{\pi}\right)^0$$$