Equations of Lines and Planes.
Level 1.
In the following series of videos we are going to cover the equations of lines and planes
in R cubed.
Before we dive in a word of caution, make sure that you are familiar with basic vector
operations such as vector addition, vector subtraction, scalar multiplication, the dot
product, and the cross product.
We will be using various vector properties in this video series.
If you are a little rusty I recommend you watch the previous videos that covered these
topics in greater detail.
Alright, we will start this video series by learning how to describe a curve in 3D space.
Recall from the three dimensional coordinate systems video series that the equation y equals
mx plus b does not describe a curve in 3D space, instead, it describes a plane, which
is a surface and not a curve.
So how do we write an equation of a curve in 3D space?
Well, we are going to need a new way of writing the equation of a curve in R cubed.
Before we talk about the equations of lines and planes we first need to briefly take a
look at vector-valued functions also known as vector functions.
We will take a look at vector functions in greater detail in a much later video for now
we will go over the basics, focusing on notation and how they can be used to describe the equation
of a curve.
In general, a function is a rule that assigns an element in the domain, to an element in
the range.
These types of functions are said to be real-valued functions.
A vector-valued function, or vector function, is simply a function whose domain is a set
of real numbers and whose range is a set of vectors.
Such functions are needed to describe curves and surfaces in space.
We can also use vector-valued functions to describe the motion of objects through space.
We usually denote vector functions by using the following notation.
A vector function will take one or more variables and return a vector.
For the purpose of describing an equation of a curve we will go over a vector function
in one variable.
We will cover two variable vector functions in a much later video.
The vector that the function generates can be a vector in whatever dimension we need
it to be.
For example, a vector function in R squared can be written as follows, and a vector function
in R cubed can be written like this.
Here, every number t in the domain of r generates a unique vector in R squared or R cubed.
For the vector function in r squared, f of t and g of t are the components of the vector
r of t, where f and g are real-valued functions called the component functions of vector r.
We can also write the vector function by using unit vectors as follows.
We use the letter t to denote the independent variable because a unit of time is used in
most applications of vector functions.
For a vector function in R cubed we include the third component in this case h of t or
k hat if we are using unit vectors.
Let’s first use a vector function in R squared to illustrate how we can determine its graph.
Consider the following vector function.
We can determine the graph of this vector function by choosing values of t and substitute
them into the vector function.
Each value of t will return a vector with components .
This vector will be a position vector that starts at the origin of a Cartesian coordinate
plane and ends at the point with coordinates (a,b).
So in order to generate the graph of this vector function we simply plug-in some values
of the variable t and then plot the point that corresponds to each position vector,
we then connect the points, each of these points will lie on the graph of the vector
function.
We can do this for other values of t, for simplicity sake let’s use the set of integers
for the values of t (keep in mind that you can use any real number for t) doing that
we obtain the following graph.
Notice that in this sketch the position vector is drawn as a gray dashed arrow for various
values of t, typically, we write the value of t above each point on the graph of a vector
function, this way we can specify the value of t that was used for each evaluation.
Taking a look at the graph we see that this vector function is nothing more than the equation
of a horizontal line y equals 1.
In a similar fashion we can describe the equation of a vertical line by using the following
vector function.
This vector function will generate various position vectors that start at the origin
and end at points that lie on the line x = 2.
You can imagine the graph of a vector function as being sketched by a position vector that
starts at the origin and moves along the curve as the value of t increases, plotting points
that lie on the curve along the way.
Keep in mind that the graphs of vector functions do not have to be straight lines.
The graphs of vector functions can generate circles, ellipses and other types of curves
and not just lines.
The following vector functions are examples of such curves.
Naturally, we can extend vector-valued functions to three dimensional space by introducing
a third component, in this case the z-component which is represented by the function h of
t in the general form of a vector function in R cubed.
Just like the vector function in R squared, a vector function in R cubed will generate
three dimensional vectors in standard form for each value of t.
The end point of each vector will trace a curve in space as t changes.
A vector function allows us to measure position, velocity, acceleration, forces and many other
quantities.
For the purpose of this video series, you should think about vector functions as position
vectors that vary with time.
These positions vectors trace the movement of the curve as time increases.
We typically draw arrows on the curve to show the direction that the end points of the position
vectors move along as t increases.
Ok and that is all you need to know about vector functions for now.
We will come back and revisit vector functions in a much later video.
Armed with this new knowledge we can now start figuring out a way to describe the equation
of a line in space.
In our next video we will do just that and derive 3 separate equations to describe a
line in 3D space.