Vector-valued functions

A function $\vec {f}: \R \to \R ^3$ can be thought of as associating to each time $t$ a vector $\vector {x(t),y(t),z(t)}$.

Consider the function $\vec {f}(t) = \vector {\cos (t),\sin (t),t}$. The projection of the point $\vec {f}(t)$ into the $(x,y)$-plane moves around the unit circle in the positive direction. The projection onto the $z$ axis moves at a constant rate in the positive direction. So we expect that $\vec {f}$ parameterizes

a straight line a circle around the $x$-axis a circle around the $y$-axis a circle around the $z$-axis a spiral around the $x$-axis a spiral around the $y$-axis a spiral around the $z$-axis

Here is the graph of $\vec {f}$:

How are vector-valued functions useful?

To get your imagination going, here are a few examples of what a function $f: \R \to \R ^3$ could represent:

• The $3$-dimensional position of a rocket in space as a function of time.
• The population of $3$ different species of bacteria found in a swimming pool as a function of the amount of chlorine in the water.
• The performance of $3$ 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 $\vec {p}$ and $\vec {q}$: $$\vecl (t) = \vec {p} + t(\vec {q}-\vec {p}).$$

Here we see vectors $\vec {p}$ and $\vec {q}$. In blue below we see the vector $t(\vec {q}-\vec {p})$ starting at point $\vec {p}$. Convince yourself that $$\vecl (t) = \vec {p} + t(\vec {q}-\vec {p})$$ draws a line.

If we know that a line passes through two points (that we’ll notate with vectors) $\vec {p}$ and $\vec {q}$, then we know that it points in the direction $\vec {v} = \vec {q} - \vec {p}$, and passes through the tip of $\vec {p}$. Hence to make a line, We write $$\vecl (t) = \vec {p}+t\vec {v}.$$ 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 $\vec {p}$, notated as the tip of a vector with its tail at the origin, and a line $$\vecl (t) = \vec {q} + t\vec {v}$$ we often want to know the distance between $\vec {p}$ and $\vecl$.

(a)

Recalling that the magnitude of a vector $|\vec {w}| =\sqrt {\vec {w}\dotp \vec {w}}$ we could attempt to minimize the function $$\mathrm {distance}(t)^2= (\vecl (t)- \vec {p})\dotp (\vecl (t)- \vec {p})$$ using the derivative. The square-root of the minimum value will be the distance.

(b)

We could compute the distance between $\proj _{\vec {v}}(\vec {p}-\vec {q})$ and $\vec {p}-\vec {q}$. This is: $$|\proj _{\vec {v}}(\vec {p}-\vec {q})- (\vec {p}-\vec {q})|$$ Checkout the diagram below:

Try to use a similar technique for points and lines in $\R ^2$:

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, $\uvec {u}$ and $\uvec {v}$, and any other vector $\vec {p}$, the vector-valued function $$\vec {f}(t) = \vec {p}+r\cdot \cos (t)\cdot \uvec {u} + r \cdot \sin (t)\cdot \uvec {v}$$ gives a circle of radius $r$, centered at the tip of $\vec {p}$, lying in the plane containing $\uvec {u}$ and $\uvec {v}$. Moreover, to produce an ellipse, we write: $$\vec {g}(t) = \vec {p}+a\cdot \cos (t)\cdot \uvec {u} + b \cdot \sin (t)\cdot \uvec {v}$$

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 $x$-component of the curve for $x$ in the equation of the surface.
• The $y$-component of the curve for $y$ in the equation of the surface.
• The $z$-component of the curve for $z$ 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:

Sometimes lines lie on surprising surfaces:

Consider the surface determined by all $x$, $y$ and $z$ such that: $$x^2+y^2=z^2+1$$ This surface looks something like:

Which of the following lines lie on the surface $x^2+y^2=z^2+1$?

Separate each line into its component functions: $x$, $y$, and $z$, and see if the equation defining the surface is valid for all $t$.

$\vector {1,0,0} + t\vector {-5,0,5}$ $\vector {1,0,0} + t\vector {0,-3,3}$ $\vector {0,1,0} + t\vector {-7,0,-7}$ $\vector {0,1,0} + t\vector {-4,4,0}$ $\vector {1/2,\sqrt {3}/2,0} + t\vector {-\sqrt {3},1,2}$ $\vector {1/2,\sqrt {3}/2,0} + t\vector {-1,\sqrt {3},2}$

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!).