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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
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
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
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
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 .
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.
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.