Question Video: Identifying the Projection of a Ray on a Straight Line | Nagwa Question Video: Identifying the Projection of a Ray on a Straight Line | Nagwa

Question Video: Identifying the Projection of a Ray on a Straight Line Mathematics • Second Year of Preparatory School

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

02:15

Video Transcript

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

In this question, we are asked to determine the result of projecting a ray onto a straight line where the two are not perpendicular. To do this, let’s start by recalling what is meant by a projection.

We first recall that the projection of a point 𝑃 onto the line between 𝐴 and 𝐵 is the point 𝑃 prime on the line between 𝐴 and 𝐵 such that the line segment 𝑃𝑃 prime is perpendicular to the line between 𝐴 and 𝐵. It is worth noting that this assumes that 𝑃 is not on the line between 𝐴 and 𝐵. If it is, then we say that the projection of 𝑃 onto the line between 𝐴 and 𝐵 is just 𝑃.

We can project more than just points by projecting every point on the object onto the line. For instance, consider the following ray from 𝑃 through 𝑄 and the line between 𝐴 and 𝐵. To project the ray onto the line, we need to project every point on the ray onto the line. We can start by projecting point 𝑃 as shown.

We can continue this process to project any point on the ray onto the line by drawing perpendicular lines between the ray and the line as shown. We need to project every point on the ray onto the line. We see that this will then be a ray. In general, it will be the ray starting at 𝑃 prime that passes through 𝑄 prime. This line of reasoning will always work provided the ray and line are not perpendicular, since this guarantees the projections of the two distinct points on the ray have distinct projections. Hence, the answer is a ray.

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