# Scalar & Vector Projections

A projection is like a shadow casted on a vector. The definition of scalar and vector projections are shown below.

### Scalar Projection

of $$\vec{a}\text{ on }\vec{b}=\vert\text{proj}\left(\vec{a}\text{ on }\vec{b}\right)\vert= \dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert}$$

of $$\vec{b}\text{ on }\vec{a}=\vert\text{proj}\left(\vec{b}\text{ on }\vec{a}\right)\vert=\dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{a}\vert}$$

### Vector Projection

Vector projection on b will result in a vector in the direction of $$\vec{b}$$.

proj$$\left(\vec{a}\text{ on }\vec{b}\right)=\dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert} \hat{b}$$

$$=\left(\dfrac{\vec{a}\cdot\vec{b}}{\vec{b}\cdot\vec{b}}\right)\vec{b}=\dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert^2}\vec{b}$$

### Vector Projection Calculator

Enter in two vectors below. Then, Click the + button to toggle between addition and subtraction.

Find the scalar and vector projections of $$\vec{u}\text{ onto }\vec{v}\text{, if }\vec{u}=(5, 6, -3)\text{ and }\vec{v}=(1, 4, 5)$$.

Find the scalar and vector projections of $$\vec{v}\text{ onto }\vec{u}$$.

If $$\vec{u}\text{ and }\vec{v}$$ are non-zero vectors, but Proj $$\vec{u}\text{ onto }\vec{v}=\vec{0}$$, what conclusion can be drawn?

If $$\vec{u}\text{ and }\vec{v}$$ are non-zero vectors, does it follow that Proj $$\vec{u}\text{ onto }\vec{v}=\vec{0}$$?

Find the projection of $$\vec{PQ}$$ onto each of the coordinate axes, where $$P$$ is the point (2, 3, 5) and $$Q$$ is the point (-1, 2, 5).

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