Vector Functions and Their Uses: Graphing and Applications

Vector Functions: The Scoop

If you read our previous post on vector calculus, you likely observed an extensive documentation of topics within the subject. And if you arrived here having just explored our post on dot products and cross products, then you are already aware of the useful computations associated with vectors. However, these computations alone are not all that useful alone. Rather, they establish the cornerstone of working with vector functions, which are highly versatile in the real world, particularly in working with data. For that reason, this post extrapolates upon vector functions, discusses their composition, and explores their usefulness in creating complex three-dimensional structures.

Vector Function Refresher

If you have not read the preceding article on vector calculus, and if you are not familiar with vector functions, let this serve as a brief refresher. Vector functions are abstractions of general functions that establish a relationship between a domain consisting of real numbers and a range consisting of vectors.

Consider a new vector ‘r’ whose domain consists of real numbers in the set ‘t’. Now, suppose that ‘r’ is constituted by three vectors defined by their own functions: ‘f’, ‘g’, and ‘h’. In this case, we may establish a relationship between ‘r’ and its component functions as follows:

r(t)=< f(t),g(t),h(t) >

Suppose that the component functions take the following inputs:

f(t)=t^3\newline g(t)=t^2 \newline h(t)=t

Then for different values of ‘t’, a new vector is defined:

\text{For } t=[1,2,3]: \newline \text{t=1: } <1,1,1 >\newline \text{t=2: } <8,4,2 >\newline \text{t=1: } <27,9,3 >

If we graph these vectors, we obtain three separate vectors pointing to different locations in a three-dimensional Cartesian plane. It appears like this:

The terminal points of these vectors coalesce to define a new entity known as a space curve. The component functions f(t), g(t) and h(t) are parametric equations of this space curve. We may observe the space curve formulated by this vector function as ‘t’ varies:

If you’d like to use such a program for yourself, check it out here. You can input the parametric equations of the vector function you are working with and it will deliver the space curve.

Vector Functions and Their Applications

Taking Limits of Vector Functions

If limits have been cause for trouble in the past, worry not. Taking limits of vector functions happens to be a rather simple operation. The limit of a vector function is simply the limit of its individual component functions, such that:

\lim_{t\to a}r(t) = < \lim_{t\to a}f(t), \lim_{t\to a}g(t), \lim_{t\to a}h(t) >

If we use our example vector function, we will see that the limit of the function is

\lim_{t\to a}r(t) = < \lim_{t\to a}t^3, \lim_{t\to a}t^2, \lim_{t\to a}t >

Let us say that a=5 and we will find:

\lim_{t\to a}r(t) = < \lim_{t\to a}t^3, \lim_{t\to a}t^2, \lim_{t\to a}t >\newline \lim_{a\to5}r(a) = < \lim_{a\to5}a^3, \lim_{a\to5}a^2, \lim_{a\to5}a >\newline \lim_{a\to5}r(a) = < 125, 25, 5 > \newline

If the limit of the function r(t) exists at some particular value of ‘t’, then we can say that the function is continuous at ‘t’ which is helpful for understanding the space curve.

Increasingly Complex Space Curves

Here we will demonstrate the working out of the space curve directly from the vector functions.

Vector Function Cylinder:

Note that a cylinder is a three dimensional structure defined by two circles serving as bounds for the base, and parallel lines connecting the two bases. The equation for a cylinder can readily be defined as:

x^2+y^2=r^2

But what of a vector function which specifies a cylinder. Is this possible? Have a look. Consider a vector function of the form:

r(t)=cos(t)+sin(t)+t

Plotting these vectors for different values of ‘t’ procures several different vectors:

t=0 : <1,0,0>\newline t=2 : <-.41,.909,2> \newline t=4 : <-.6536, -.7568, 4>\newline t=6 : <.96, -.2794, 6> \newline t=8 : <-.1455, .9893, 8>

If we were to graph these, we would see several vectors of increasing magnitude, whose terminal points revolve around the z-axis. However, there is not much variation in their x/y-displacements:

This alone, however, provides little insight. Let us graph the space curve:

This particular curve is known as a helix, and it forms the basis of mathematically modeling DNA.

Finding A Vector Function Defining the Intersection Between Two Structures:

Suppose we have a cylinder defined by the function:

x^2+y^2=1

In addition, suppose we have a plane defined by the function:

y+z=2

If these were overlayed atop each other, what would be the vector function that defined this intersection? First, let us consider what this interaction looks like in three dimensions:

Here we observe a three dimensional interaction between the cylinder and the plane. So, how can we determine the vector function that describes their intersection?

Well, we know from our previous example that the vector function which defines a cylinder has the following form:

r(t)=< cos(t), sin(t), t >

Now, from the equation of the plane, we can modify it to be:

z=2-y

The cylinder shows us that y=sin(t), so let’s substitute this into the plane equation to give ourselves:

z=2-sin(t)

Now, we can plug this in to the vector function of the cylinder to give ourselves:

r(t)=< cos(t), sin(t), 2-sin(t) >

Graphing this function provides us with the following vector:

More On Vector Functions

Associated with the fundamental attributes of vector functions are the processes of differentiation and integration of these relationships. Doing so allows for more sophisticated analysis of a computational system. A future post intends to address in depth the processes of vector differentiation and integration, as well as explore their uses in computing in arc length and curvature. If you’ve found this to be a useful resource, don’t hesitate to check out other posts which explore these topics in great detail.

  1. General Vector Calculus
  2. Dot Product and Cross Product
  3. Vector Function (current)
  4. Vector Differentiation and Integration
  5. Arc Length and Arc Curvature

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