Vector Projection Calculator

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

Related calculator: Scalar Projection Calculator

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Calculate the vector projection of $\mathbf{\vec{v}} = \left(-4, 2, 7\right)$ onto $\mathbf{\vec{u}} = \left(3, 1, 2\right)$.

Solution

The vector projection is given by $\text{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}^{2}} \mathbf{\vec{u}}.$

$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 4$ (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 $\text{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{4}{\left(\sqrt{14}\right)^{2}}\cdot \left(3, 1, 2\right) = \frac{2}{7}\cdot \left(3, 1, 2\right) = \left(\frac{6}{7}, \frac{2}{7}, \frac{4}{7}\right)$ (for steps, see vector scalar multiplication calculator).

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