# Vector Projection Calculator

The calculator will find the vector projection of one vector onto another, with steps shown.

Related calculator: Scalar Projection Calculator

$$\mathbf{\vec{u}}$$$: ( , , ) $$\mathbf{\vec{v}}$$$: (
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Hint: if you have two-dimensional vectors, set the third coordinates equal to or leave them empty.

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Calculate the vector projection of $$\mathbf{\vec{v}} = \left(-4, 2, 7\right)$$$onto $$\mathbf{\vec{u}} = \left(3, 1, 2\right)$$$.
The vector projection is given as $$\text{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{\mathbf{\vec{u}}\cdot \mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{u}\right\rvert}^{2}} \mathbf{\vec{u}}.$$$$$\mathbf{\vec{u}}\cdot \mathbf{\vec{v}} = 4$$$ (for steps, see dot product calculator).
$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{14}$$$(for steps, see vector magnitude calculator). Thus, the vector projection is $$\text{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{4}{\left(\sqrt{14}\right)^{2}} \left(3, 1, 2\right) = \left(\frac{6}{7}, \frac{2}{7}, \frac{4}{7}\right).$$$
The vector projection is $$\left(\frac{6}{7}, \frac{2}{7}, \frac{4}{7}\right)\approx \left(0.857142857142857, 0.285714285714286, 0.571428571428571\right).$$\$A