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Projections

There are two types of projections that you need to be aware of at this stage: scalar projections and vector projections.

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# Projections

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There are two types of projections that you need to be aware of at this stage: scalar projections and vector projections.

Scalar projection simply gives the length in a particular direction. The result of this is a scalar that quantifies this amount. On the other hand, vector projection 'projects' the length of one vector in the direction of another. One way to think of it is as if the 'shadow' of a vector is cast over the top of another vector.

## 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.

The scalar projection of a vector $\mathbf{x}$ onto unit vector $\stackrel{^}{\mathbf{u}}$ is the scalar given by the dot product: $\mathbf{x}·\stackrel{^}{\mathbf{u}}$ or

Figure 1

Find the scalar projection of vector $\stackrel{\to }{\mathrm{OA}}$ in the horizontal direction.

Solution

The 'horizontal direction' is along the x-axis, therefore the unit vector we will be using is $\stackrel{^}{\mathbf{u}}=\left[\begin{array}{c}1\\ 0\end{array}\right]$. Vector $\stackrel{\to }{\mathrm{OA}}$ in vector notation is $\left[\begin{array}{c}3\\ 4\end{array}\right]$.

Intuitively, you may have already realized that the scalar projection must be 3 since by definition $\stackrel{\to }{\mathrm{OA}}$ is comprised of 3 units in the horizontal direction (and 4 vertically).

We can show this using the dot product too:

Therefore, the scalar projection of $\stackrel{\to }{\mathrm{OA}}$ in the horizontal direction is equal to 3.

## 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 ${\mathrm{Proj}}_{\mathbit{c}}\left(\mathbit{a}\right)$. We know that the vector $\underset{BA}{\to }$must be equal to $\mathbit{a}-\mathbit{b}=\mathbit{a}-{\mathrm{Proj}}_{\mathbit{c}}\left(\mathbit{a}\right)$, therefore ${\mathrm{Proj}}_{\mathbit{c}}\left(\mathbit{a}\right)$ is such that $\mathbit{a}-{\mathrm{Proj}}_{\mathbit{c}}\left(\mathbit{a}\right)$ 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${\mathrm{Proj}}_{\mathbit{v}}\left(\mathbit{a}\right)$ is in the direction of L, we can write it as a scalar multiple of vector v.

${\mathrm{Proj}}_{\mathbit{v}}\left(\mathbit{a}\right)$$c\mathbf{v}$
• v is some vector in the 2D space

• c is some constant

Using the dot product:

$\left(\mathbf{a}-{\mathrm{Proj}}_{\mathrm{L}}\left(\mathbf{a}\right)\right)·\mathbf{v}=\left(\mathbf{a}-\mathrm{c}\mathbf{v}\right)·\mathbf{v}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbf{a}·\mathbf{v}-\mathrm{c}\mathbf{v}·\mathbf{v}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbf{a}·\mathbf{v}=\mathrm{c}\mathbf{v}·\mathbf{v}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{c}=\frac{\mathbf{a}·\mathbf{v}}{\mathbf{v}·\mathbf{v}}$

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

The vector projection of vector a onto vector v is ${\mathrm{Proj}}_{\mathrm{L}}\left(\mathbf{a}\right)=\frac{\mathbf{a}·\mathbf{v}}{{\left|\mathbf{v}\right|}^{2}}\mathbf{v}$.

We have vector $\mathbf{x}=\left[\begin{array}{c}6\\ 2\end{array}\right]$ and vector $\mathbf{v}=\left[\begin{array}{c}7\\ -6\end{array}\right]$. Find the projection of x onto v.

Using the formula: ${\mathrm{Proj}}_{\mathbf{v}}\left(\mathbit{x}\right)=\frac{\mathbf{x}·\mathbf{v}}{{\left|\mathbf{v}\right|}^{2}}\mathbf{v}$

${\left|\mathbf{v}\right|}^{2}={\left(\sqrt{{7}^{2}+{\left(-6\right)}^{2}}\right)}^{2}=85$

$\mathbf{x}·\mathbf{v}=\left[\begin{array}{c}6\\ 2\end{array}\right]·\left[\begin{array}{c}7\\ -6\end{array}\right]=42-12=30$

Therefore, ${\mathrm{Proj}}_{\mathbf{v}}\left(\mathbit{x}\right)=\frac{30}{85}\mathbf{v}=\frac{6}{17}\left[\begin{array}{c}7\\ -6\end{array}\right]$.

${\mathrm{Proj}}_{\mathbf{v}}\left(\mathbf{x}\right)$ is represented by the blue arrow in the diagram below.

Figure 4

## Projections - Key takeaways

• Scalar projection gives the length in which a vector is in a given direction
• To find the scalar projection, use the dot product of a vector with a unit vector in the direction in question
• The formula for scalar projection of x onto direction u is $\mathbf{x}·\stackrel{^}{\mathbf{u}}$
• Vector projection projects the length of one vector into the direction of another
• To find the projection of vector a onto L, ${\mathrm{Proj}}_{\mathrm{L}}\left(\mathbf{a}\right)$ is such that $\mathbf{a}-{\mathrm{Proj}}_{\mathrm{L}}\left(\mathbf{a}\right)$ is orthogonal to line L
• The projection of vector a onto line L of vector v is ${\mathrm{Proj}}_{\mathrm{L}}\left(\mathbf{a}\right)=\frac{\mathbf{a}·\mathbf{v}}{{\left|\mathbf{v}\right|}^{2}}\mathbf{v}$.

Scalar projection gives the scalar length in a particular direction.  On the other hand, vector projection 'projects' the length of one vector in the direction of another.

Two examples of projection are scalar projection and vector projection.

Scalar projection is given by the dot product.

Both scalar projection and vector projection are done by applying their respective formula.

Projections in geometry do not fall under the scope of the article. Nevertheless, they are spherical, cylindrical, conical, and azimuthal projections.

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