Projections

Scalar projection

We will start with the simplest to understand conceptually. The scalar projection of a vector finds how much, as a scalar, of the vector is in a particular direction. It is found using the dot product of the vector with the unit vector in the direction in question.

Vector projection

A vector projection is the projection of one vector onto another. It takes the length of one vector and projects it in the direction of another, creating a new vector with the direction of the second.

Figure 2

Figure 3

Vector b is known as the projection of a in the x-axis direction (figure 2). If you look at figure 3, vector c is a vector in the x-axis direction, so b is also the projection of a in the direction of vector c.

In mathematical notation, this is written as ${\mathrm{Proj}}_{\mathit{c}}\left(\mathit{a}\right)$. We know that the vector $\underset{BA}{\to }$must be equal to $\mathit{a}-\mathit{b}=\mathit{a}-{\mathrm{Proj}}_{\mathit{c}}\left(\mathit{a}\right)$, therefore ${\mathrm{Proj}}_{\mathit{c}}\left(\mathit{a}\right)$ is such that $\mathit{a}-{\mathrm{Proj}}_{\mathit{c}}\left(\mathit{a}\right)$ is orthogonal to vector c.

This orthogonality is a property that is essential to finding the projection of a onto L. Remember the 'dot product'? Since $\underset{BA}{\to }$is orthogonal to line L, the 'dot product' of the two must be equal to zero. Using this information, we can derive a formula for the vector projection of a in the direction of b.

Deriving the projection vector formula

To understand fully what the projection vector does, it may be useful to see how to derive it. First, let's consider a vector v lying on line L. Since${\mathrm{Proj}}_{\mathit{v}}\left(\mathit{a}\right)$ is in the direction of L, we can write it as a scalar multiple of vector v.

• v is some vector in the 2D space

• c is some constant

Using the dot product:

$$\begin{gathered} (\mathbf{a}-{\mathrm{Proj}}_{\mathrm{L}}(\mathbf{a}\left)\right)·\mathbf{v}=(\mathbf{a}-\mathrm{c}\mathbf{v})·\mathbf{v}=0 \\ \mathbf{a}·\mathbf{v}-\mathrm{c}\mathbf{v}·\mathbf{v}=0 \\ \mathbf{a}·\mathbf{v}=\mathrm{c}\mathbf{v}·\mathbf{v} \\ \mathrm{c}=\frac{\mathbf{a}·\mathbf{v}}{\mathbf{v}·\mathbf{v}} \end{gathered}$$

Since the dot product of a vector with itself is equal to the length of that vector squared, we get:

$\mathrm{c}=\frac{\mathbf{a}·\mathbf{v}}{|v{|}^{\mathrm{2}}}$.

And so, ${\mathrm{Proj}}_{\mathrm{L}}\left(\mathbf{a}\right)=\frac{\mathbf{a}·\mathbf{v}}{{\left|\mathbf{v}\right|}^2}\mathbf{v}$.