# Vector projection of $\left\langle 4, -2, -4\right\rangle$ onto $\left\langle 1, 2, -2\right\rangle$

The calculator will find the vector projection of the vector $\left\langle 4, -2, -4\right\rangle$ onto the vector $\left\langle 1, 2, -2\right\rangle$, with steps shown.

Related calculator: Scalar Projection Calculator

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Calculate the vector projection of $\mathbf{\vec{v}} = \left\langle 4, -2, -4\right\rangle$ onto $\mathbf{\vec{u}} = \left\langle 1, 2, -2\right\rangle$.

### Solution

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

$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = 8$ (for steps, see dot product calculator).

$\mathbf{\left\lvert\vec{u}\right\rvert} = 3$ (for steps, see vector magnitude calculator).

Thus, the vector projection is $\text{proj}_{\mathbf{\vec{u}}}\left(\mathbf{\vec{v}}\right) = \frac{8}{3^{2}}\cdot \left\langle 1, 2, -2\right\rangle = \frac{8}{9}\cdot \left\langle 1, 2, -2\right\rangle = \left\langle \frac{8}{9}, \frac{16}{9}, - \frac{16}{9}\right\rangle$ (for steps, see vector scalar multiplication calculator).

The vector projection is $\left\langle \frac{8}{9}, \frac{16}{9}, - \frac{16}{9}\right\rangle\approx \left\langle 0.888888888888889, 1.777777777777778, -1.777777777777778\right\rangle.$A