Vector-valued functions are parameterized curves.

### Vector-valued functions

A function can be thought of as associating to each time a vector .

Vector-valued functions simply map numbers to lists of numbers, that we interpret as vectors:

#### 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 :

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 . When we will know that the line passes through the tip of , and points in a direction . Then we write Play around with the interactive below to see if you get the idea:

There are an infinite number of ways to parameterize the same line. Try your hand at the following puzzlers:

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 .

- (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:

*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

Try to use a similar technique for points and lines in :

However, depending on the question, you might want to think before blindly applying formulas. Try your hand at this last question:

### 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:

**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.

### 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:

Which of the following lines are on both of these planes?

Sometimes lines lie on surprising surfaces:

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!).