A line in the Cartesian Plane is a set of Ordered Pairs.
Lines contain information, but we do not want to inspect each individual
point because every line contains an infinite number of points. Fortunately,
a line has properties which give us access to its information.
Distance is a property a line inherits from the Cartesian Plane. Other properties
are derived from this fact. Distance is the static difference in position
between two points. If something moves from one point to another there is
a net change in its coordinates. This net change is the increment we are
going to use to define properties of a line.
The coordinate increments between P and Q are:
Δx = x2 - x1
Δy = y2 - y1
Where Δ, the Greek letter delta, means change or difference.
Since we can calculate Δx and Δy from any two points
on a line we can see how y changes with respect to x. This is called the
slope and it is the same at every point on a straight line.
m =
y2 - y1
=
Δy
=
rise
x2 - x1
Δx
run
where the slope is defined by m.
I need to say one word about Trigonometry here,
"soh-cah-toa".
Do
not be alarmed by the tangent of Φ. What is important for right now
is that,
m =
Δy
Δx
Mathematics has many branches. From Set Theory we can build
a coordinate Geometry and though this page is about Lines it looks like
Algebra to me. Now Trigonometry has snuck in. See, Mathematics is a language
built on ideas and experience to express and explore concepts in some kind
of reality. Anyway, I will do my best to wander on as a direct path as I
can to understand General Relativity.
At this time we can say two things about the
slope of a line. If the line is horizontal, it is parallel to the x-axis,
y is constant so Δy = 0 then the slope of a horizontal line is zero.
If the line is vertical, it is parallel to the y-axis, x is constant so
Δx = 0 then the slope of a vertical it undetermined.
In the picture of line L above you can increase Δy and decrease Δx
to bring line L into a vertical position. In the process the slope m become
huge. Even if you leave Δy alone and just decrease Δx line L still
approaches the vertical and the slope m becomes huge.
The line L goes up and to the right. It has a positive slope. If line L
was redrawn going down and to the right or simply change the sign of Δy
to negative the slope becomes negative.
By
repeating the experiments using line L as it is now we clarify the concept
that decrease means approaching zero and huge can be either negative or
positive. Lastly, we have determined the range of m is -
< m < .
Parallel and Perpendicular Line
The proof about m concerning || and
lines
If
m1 = m2 and L1 does not have a point in
common with L2, then L1 and L2 are parallel.
comes from Euclidean Geometry, but to indulge in that proof
right now will take us far afield from the tools we are currently collecting.
Euclidean Geometry is based on five axioms. The first four dictate our Basic
Geometry. Adding Euclid's Fifth Axiom defines Euclidean Geometry.
If
neither L1 and L2 are vertical line and m1m2
= -1, then L1 and L2 are perpendicular.
We are going to use these statements about parallel and perpendicular lines
as if they are simply facts.
For parallel lines, just look at the picture. For perpendicular lines, lets
develop our own intuition.
It is possible to draw one and only one straight line from any point
to any point.
From each end of a finite straight line it is possible to produce
it continuously in a straight line by a greater than any length.
It is possible to describe one and only one circle any center and
radius.
All right angles are equal to one another.
(Euclid's Fifth Axiom) Through a given point not on a given straight
line, and not on that line produced, no more than one parallel straight
line can be dra
Imagine yourself in a city where the streets are laid out
in a grid. They run north-south and east-west. Main Street is an east-west
street. Standing in the center of the city you find yourself at City Hall
on Main Street. Being a bit hungry, you walk five blocks east and twelve
blocks north to arrive at a Hot Dog Stand. You are now thirteen blocks north-northeast,
as the crow flies, from where you started. After eating a hot dog, you turn
yourself into a crow and fly south-southwest back to City Hall.
Please do not draw on the monitor if you are using the compass card for
directions.
Once back at City Hall you turn back into a human and find
that you are feeling thirsty. This time you walk twelve blocks east and
five blocks south to get a drink at the only cold Drinking Fountain in the
city. Realizing that it is almost time for your meeting at City hall, you
do the math. You walked seventeen blocks to arrive at the Drinking Fountain,
thirteen blocks east-southeast of City Hall. Transforming once again into
a crow, you set off west-northwest towards City Hall.
That's the basic idea. Any two lines that have slopes such that m1m2
= -1 are perpendicular.
The equation of a line determines the set of coordinates
contained by that line. The general form of the equation for horizontal
lines is x = n and for vertical lines is y = n where n . The equation of a
line can be defined for non-vertical straight lines and comes in three forms:
the Point-Slope Equation, the Slope-Intercept Equation and the General Linear
Equation.
All that is required to construct the Point-Slope Equation
are two points on the line under investigation. After your walk in Grid
City you are an expert at finding the slope of a line using,
m =
y2 - y1
x2 - x1
Multiplying both sides of the equation by (x2 - x1)
results in (y2 - y1) = m(x2 - x1).
Since m is a constant value for every point on the line we can keep (x1,
y1) and solve for any other coordinate (x, y) on the line.
The Point-Slope Equation is
(y - y1) = m(x - x1).
The Point-Slope Equation is not a bad looking equation and
of course for a specific line m, y1 and x1 will have
specific numbers. Once m, y1 and x1 are replaced by
numbers we can solve the Point-Slope Equation for any point on the line.
A useful point to find is the y-intercept which is where (x, y) will be
(0, b). So using the Point-Slope Equation again the y-intercept has numeric
values.
The y-intercept is now a known point on the line of interest.
Use this coordinate to replace y1 and x1 to get
(y - b) = m(x - 0)
which simplifies to
y = mx + b
and gets the name Slope-Intercept Equation.
The General Linear Equation can be arrived at through a rearrangement
of the equations just discussed. It will prove useful and it looks like
this:
Do
not be alarmed by the tangent of Φ. What is important for right now
is that,
By
repeating the experiments using line L as it is now we clarify the concept
that decrease means approaching zero and huge can be either negative or
positive. Lastly, we have determined the range of m is -
< m < 
lines
If
m1 = m2 and L1 does not have a point in
common with L2, then L1 and L2 are parallel.
If
neither L1 and L2 are vertical line and m1m2
= -1, then L1 and L2 are perpendicular.
. The equation of a
line can be defined for non-vertical straight lines and comes in three forms:
the Point-Slope Equation, the Slope-Intercept Equation and the General Linear
Equation.