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.
