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Mathematical Expression Editor
Vector-valued functions are parameterized curves.
Vector-valued functions
A function can be thought of as associating to each time a vector .
A
vector-valued function maps real numbers to vectors in .
Vector-valued functions simply map numbers to lists of numbers, that we interpret
as vectors:
Placing the tail of the vector at the origin, its head will sweep out a curve
parameterized by . Below we see a plot of the vector-valued function:
Use the slider to see how the vector-valued function is “drawn” by the tip of the
vector
Consider the function . The projection of the point into the -plane moves around the
unit circle in the positive direction. The projection onto the axis moves at a constant
rate in the positive direction. So we expect that parameterizes
a straight linea
circle around the -axisa circle around the -axisa circle around the -axisa
spiral around the -axisa spiral around the -axisa spiral around the -axis
Here is the graph of :
How are vector-valued functions useful?
To get your imagination going, here are a few examples of what a function could
represent:
The -dimensional position of a rocket in space as a function of time.
The population of different species of bacteria found in a swimming pool
as a function of the amount of chlorine in the water.
The performance of different stocks as a function of time.
The trunk width, height, and canopy radius of a tree as a function of time.
The average temperature, humidity, and air pressure at a given latitude
as a function of that latitude.
The RGB color of a single pixel of a LCD screen varying over time.
Of the examples above, perhaps “position in space” is the best mental model to use
to help you understand vector-valued functions.
Lines in space
It is easy to create a vector-valued function that passes through two points and :
What is the value of ?
is unknowable
What is the value of ?
is unknowable
What value of gives the midpoint of the tips of vectors and ?
Here we see vectors and . In blue below we see the vector starting at point .
Convince yourself that draws a line.
If we know that a line passes through two points (that we’ll notate with vectors) and
, then we know that it points in the direction , and passes through the tip of . Hence
to make a line, We write Play around with the interactive below to see if you get the
idea:
Using the ideas above, find an expression in terms of parameterizing the line
passing through when , and when .
The line passes through and points in the direction
Let be a line that passes through the points and . What are the components of ?
Let , then .
There are an infinite number of ways to parameterize the same line. Try your hand at
the following puzzlers:
Compare and contrast the curves and .
They parameterize different lines.They
parameterize the same line, but moves “twice as fast” as .They parameterize the
same line, but moves “twice as fast” as .These are the same function!
Note, both lines start at the same point when .
We can rewrite and as:
We can further rewrite as:
Compare and contrast the curves and .
They parameterize different lines.They
parameterize the same line, but moves in the opposite direction compared with .They parameterize the same line, but moves “twice as fast” as .These are the
same function!
Note both lines start at the same point when .
We can rewrite and as:
We can further rewrite as:
We can use these ideas to parameterize any line in space. However, our
parameterizations will not be unique as there are infinitely many different ways to
parameterize the same line. Some parameterizations may “move faster” than others,
or in the opposite direction, or even at uneven rates!
Distance between a point and a line
Given a point , notated as the tip of a vector with its tail at the origin, and a line we
often want to know the distance between and .
This distance is the length of the shortest path from to the line . How do we find
this distance? Well:
(a)
Recalling that the magnitude of a vector we could attempt to minimize
the function using the derivative. The square-root of the minimum value
will be the distance.
(b)
We could compute the distance between and . This is: Checkout the
diagram below:
However, both of these methods are somewhat involved. Perhaps the quickest method
for determining the distance between a point and a line is by using the cross product.
Since the cross product is only defined in , we need -dimensional vectors. If we
consider the vector , we see by the definition of sine
that the distance we are looking for is given by However, so we see that Try your
hand at it by answering the following questions:
What is the distance between the point and the line that passes through the origin
and ?
Try to use a similar technique for points and lines in :
What is the distance between the point and the line that passes through the points
and ?
To use the cross product, make these points -dimensional by adding a
-component of to each point.
In this case .
However, depending on the question, you might want to think before blindly applying
formulas. Try your hand at this last question:
Consider the line: What point on is nearest to the point ?
Here, we are not asking
for the distance, we are asking for the nearest point.
It will be easiest to use an orthogonal projection to answer this question.
The point on closest to is:
Circles and ellipses
Given two orthogonal unit vectors, and , and any other vector , the vector-valued
function gives a circle of radius , centered at the tip of , lying in the plane containing
and . Moreover, to produce an ellipse, we write:
Given an ellipse, the major axis of
an ellipse is its longest diameter, and the minor axis is its smallest diameter. The
semi-major axis is half of the major axis, and the semi-minor axis is half of the
minor axis. Given an ellipse of the form where , is the semi-major axis and is the
semi-minor axis.
Let’s see an example.
Give a vector-valued formula for an ellipse that is drawn in the -plane centered
at the point whose semi-major axis is on a line parallel to the -axis, and
whose semi-minor axis is on a line parallel to the -axis.
There are actually
infinitely many solutions to this problem, though we’ll just give one. Write:
Can you find a vector-valued formula for a circle of radius in the plane centered at
? A circle we seek is:
Lines and curves embedded in surfaces
Curves can lie on surfaces. Typically, the surface is defined implicitly, and the curve
is a vector-valued function. To check if the curve lies on the surface, break the curve
into components and substitute:
The -component of the curve for in the equation of the surface.
The -component of the curve for in the equation of the surface.
The -component of the curve for in the equation of the surface.
If the equation defining the surfaces holds after the substitution, the curve lies on the
surface. Try your hand at these puzzles:
Consider the plane: Which of the following lines are on this plane?
Separate each
line into its component functions: , , and , and see if the equation defining the surface
is valid for all .
Consider the planes:
Which of the following lines are on both of these planes?
Separate each line into its component functions: , , and , and see if the equation
defining each surface is valid for all .
Sometimes lines lie on surprising surfaces:
Consider the surface determined by all , and such that: This surface looks
something like:
Which of the following lines lie on the surface ?
Separate each line into its
component functions: , , and , and see if the equation defining the surface is valid for
all .
Though their formulation may be more complex, a vector-valued function that
produces a curve is no different from that which produces a line (a line is a special
type of curve!).
Consider the unit sphere: Which of the following curves lie on this sphere?
Separate
each curve into its component functions: , , and , and see if the equation defining the
surface is valid for all .
Consider the surface determined by all , and such that: Which of the
following curves lie on the surface ?
Separate each curve into its component
functions: , , and , and see if the equation defining the surface is valid for all
.