Scalar Projection Calculator

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

Related calculator: Vector Projection Calculator

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Calculate the scalar projection of $\mathbf{\vec{v}} = \left(7, 0, 5\right)$ onto $\mathbf{\vec{u}} = \left(1, -3, -4\right)$.

Solution

The scalar projection is given by $\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}}$.

$\mathbf{\vec{v}}\cdot \mathbf{\vec{u}} = -13$ (for steps, see dot product calculator).

$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{26}$ (for steps, see vector magnitude calculator).

Thus, the scalar projection is $\frac{\mathbf{\vec{v}}\cdot \mathbf{\vec{u}}}{\mathbf{\left\lvert\vec{u}\right\rvert}} = \frac{-13}{\sqrt{26}} = - \frac{\sqrt{26}}{2}.$

The scalar projection is $- \frac{\sqrt{26}}{2}\approx -2.549509756796392$A.