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 .

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.

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.)


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.