Lesson Explainer: Projections

In this explainer, we will learn how to find the projection of a point, a line segment, a ray, or a line on another line and find the length of the projection.

In general terms, we can think of a projection as the shadow casted by the object. This can be a useful visualization that has many real-world applications. For example, consider a ball that is thrown in an arc. We can project the ball onto the ground by considering its shadow with respect to the sun. Of course, the position of the sun will affect where the shadow is. We are only interested in vertical projection, which means the sun will be directly above in our example. We get the following.

It is useful to consider the projection of the ball on the ground in this case since we are often interested in the horizontal distance traveled by the ball rather than the total distance traveled. In this case, we can measure the distance traveled by its projection to the ground. In other words, the horizontal distance traveled by the ball is the same as the horizontal distance traveled by its shadow. We can formalize this definition for projecting a point onto a line as follows.

Definition: Projection of a Point onto a Straight Line

The projection of a point onto a straight line is the point such that .

If , then we say the projection of onto is just .

With this definition in mind, we can think of finding the area of a triangle in another way. Recall that the area of a triangle is half the length of its base multiplied by its perpendicular height. If we consider a sketch where we choose as the base, we have the following.

We see that meets at right angles, so we can say that is the projection of onto . Since is the perpendicular height of the triangle, we can also think about this as the length between and its projection onto the line through the base.

Let’s now see an example of finding the projection of a point onto a line.

Example 1: Finding the Projection of a Point on a Line

What is the projection of on ?

Answer

We recall that the projection of a point onto a straight line is the point on the line such that . We can see in the diagram that and that .

Hence, the projection of on is the point .

We can extend the idea of the projection of a single point onto a line by considering how we could project a line segment onto a line. To go back to our ball example, we may want to know the horizontal distance traveled between two times. In this case, we would want to project both the start and endpoints of the ball’s trajectory onto the ground as shown.

In this scenario, we are interested in the line segment between the projections of the endpoints. This idea is how we project a line segment onto a straight line. We can describe this formally as follows.

Definition: Projection of a Line Segment onto a Straight Line

The projection of a line segment onto a straight line is the line segment , where and are the projections of and onto respectively.

It is worth noting that if , then and will be the same point. In this case we say that the projection is the single point .

It is worth noting that, for any point , if we name its projection onto as , then we must have .

Let’s now see an example of finding the projection of a line segment onto a given straight line.

Example 2: Finding the Projection of a Line Segment on a Line

What is the projection of on ?

Answer

We begin by recalling that the projection of a line segment onto a straight line is the line segment between the projections of the endpoints. Therefore, to project onto we first need to separately project and onto .

We can then recall that the projection of a point onto a straight line is the point on the line such that .

We can see in the diagram that and that . So, is the projection of onto . Finally, we recall that, since , its projection onto is itself.

Hence, the projection of on is .

In our next example, we will consider the projection of a perpendicular line segment onto a straight line.

Example 3: Finding the Projection of a Line Segment Perpendicular to a Line

What is the projection of on ?

Answer

We begin by recalling that the projection of a line segment onto a straight line is the line segment between the projections of the endpoints. Therefore, to project onto , we first need to separately project and onto .

First, we can see that , hence its projection onto this line is itself. Next, we see in the diagram that , so the projection of onto this line is also .

Therefore, since the line segment is just the point , we have shown that projection of on is the point .

In our next example, we will determine the length of the projection of a line segment.

Example 4: Finding the Length of the Projection of a Line Segment on a Line

Given that , , and , calculate the length of the projection of on .

Answer

We first note that , so its projection onto this line is unchanged. To project onto , we need to find a line perpendicular to that passes through . We can do this by noting that and is a transversal of these parallel lines. Hence, angles and are alternate, so

Therefore, the projection of onto is , so we need to determine . This gives us the following.

We can determine this length by applying the Pythagorean theorem twice. First, on triangle , we have

Substituting and into the equation gives us

We can then rearrange this to get

Taking the square root of both sides of the equation where we note that is a length and so is nonnegative gives us

We can add this to the diagram.

We can now determine the length of by applying the Pythagorean theorem to ; we have

Substituting and into the equation gives us

We can then rearrange the equation and evaluate it to get

Finally, we take square roots of both sides of the equation to get

A useful fact to note is that the length of the projection of a line segment is always less than or equal to the length of the original line segment. We can use this as a quick check for our answers. We can also prove this result by considering a projection.

We can show that by translating line vertically to form a right triangle.

We can now see that is the hypotenuse of this right triangle, so . It is worth noting that we assume that one of or does not lie on ; otherwise, the projection leaves the line segment unchanged. This gives the following result.

Property: Projection Length of a Line Segment

The length of the projection of a line segment is always less than or equal to the length of the original line segment.

Let’s see an example of using this property after we find the length of the projection of a line segment to check our answer.

Example 5: Finding the Length of the Projection of a Line Segment on a Line

Find the length of the projection of on .

Answer

To project these points onto the line, we need to find the lines through and that are perpendicular to . We note that and , so is the projection of onto . If we draw the perpendicular from to , we note that this is the perpendicular height of ; we will call the point on the base as shown.

Hence, the projection of onto is .

We can then note that is an isosceles triangle and so its perpendicular from vertex to the side will bisect . Then, since , we must have that .

We can determine the length of by applying the Pythagorean theorem to . We have

Substituting and yields

We can then rearrange and evaluate to get

Taking the square root of both sides of the equation where we note that is a length and so is nonnegative gives us

We can then determine the length of as half of . We get

Finally,

There is one final type of projection, which is the projection of a ray onto a line. We recall that a ray is a straight line with a given direction and start point but no endpoint. We can project a ray onto a line by projecting its start point and one other point on the ray and then taking the ray starting at the projected start point and going through the other projected point. We can write this formally as follows.

Definition: Projection of a Ray onto a Line Segment

The projection of a ray onto a straight line is the ray where and are the projections of and onto respectively.

It is worth noting that, if is perpendicular to , then and will be the same point. In this case, we call the projection the single point .

Finally, it is worth noting that we can also project a line onto another line in the same way. In this case, if the two lines are not perpendicular, then the projection will be the second line. If the two lines are perpendicular, then the projection will just be the point of intersection.

Let’s see an example of using this definition.

Example 6: Recognizing the Projection of a Ray onto a Line

What is the result of projecting a ray onto a straight line, given that the two are not perpendicular?

Answer

We recall that the projection of a ray onto a straight line is the ray where and are the projections of and onto respectively. Since is not perpendicular to , we know that and will have distinct projections onto . If we call these projections and , respectively, then we have the following.

Therefore, the projection of onto is the ray .

Hence, the answer is a ray.

Let’s finish by recapping some of the important points from this explainer.

Key Points

The projection of a point onto a straight line is the point such that . Where we note that, if , then we say the projection of onto is just .
The projection of a line segment onto a straight line is the line segment , where and are the projections of and onto respectively. Where we note if the line segment and straight line are perpendicular, then both endpoints project to the same point and so we say the projection is a single point.
The length of the projection of a line segment is always less than or equal to the length of the original line segment.
The projection of a ray onto a straight line is the ray where and are the projections of and onto respectively. Where we note that, if is perpendicular to , then and will be the same point. In this case, we call the projection the single point .