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Mathematical Expression Editor

We find the projection of a vector onto a given non-zero vector, and find the distance
between a point and a line.

VEC-0070: Orthogonal Projections

Given a line and a vector emanating from a point on , it is sometimes convenient to
express as the sum of a vector , parallel to , and a vector , perpendicular to
.

Suppose is a direction vector for . Then for some scalar . Our goal is to find .

We conclude that
and

The vector is called the projection of onto . In our discussion, is a direction vector
for line . So, we can also say that is the projection of onto .

To find , observe that .

Let be a vector, and let be a non-zero vector. The projection of onto is given
by

Find the projection of , shown below, onto the line given by .

We begin by finding vectors and . The tail of is located at , and the head of is at .
Using the “head-tail” formula we get
The direction vector for the line is
We find that and . Thus

Distance from a Point to a Line

The shortest distance from a point to a line is the length of the perpendicular line
segment dropped from the point to the line. Vector projection formula will help us
find the length of such a perpendicular.

Let be a point in . Suppose line is given by parametric equations

Find the distance from to .

We will first construct a vector by picking an
arbitrary point on to be the tail of and using point as the head of . An
easy point to choose on line is the point that corresponds to . Now we
have

The line has a direction vector

We will now find the projection of onto

Next, we find .

Finally, to find the distance between point and line , we find the magnitude of
.

Practice Problems

Find .

If and then

If and then

Find the projection of vector onto line . (If entering answers in decimal form, round
to the nearest one hundredth.)

Answer:

Find the distance between point and line .

Answer: .

Show that does not depend on the length of by proving that for . What does this
result mean geometrically? Illustrate your response with a diagram.

Find the radius of a circle centered at if the line is tangent to the circle. Enter
your response as a fraction.

Answer:
The graph below shows the line together with a circle of radius . Change the value
of to the radius you have found to visualize the correct answer.