two points on the line, and (x,y) are the coordinates of
an arbitrary point on the line, then we have:
=
This provides the “two-point form” equation of the line.
For example, the equation of the line passing through
the points (2,3) and (–1,5) is = = .
The quantity is the slope mof the line.
Thus one can rewrite the two-point form of the equation
as . Rearranging yields:
y– b1= m(x– a1)
This is called the “point-slope form” equation of the
line. For example, the equation of a line of slope 4 that
passes through the point (5,7) is simply y– 7 = 4(x– 5).
Working with this form of equation is useful if the slope
of the line is already specified.
Rearranging the point-slope equation yields y= mx
+ (ma1+ b1). Denoting the constant ma1+ b1simply as
byields the equation:
y= mx + b
Noting that if x= 0, then we have y= b. This shows
that the constant bis the y-intercept of the line. For
this reason, the above equation is called the “slope-
intercept form” equation of the line. Thus, for exam-
ple, the equation of a line with slope –1 crossing the
y-axis at position 3 is y= –x+ 3.
One disadvantage of the slope-intercept form is
that it does not allow one to write down the equation
of a vertical line, that is, one that does not intercept the
y-axis at all. Returning to the two-point form and
cross-multiplying yields the equation (a2– a1)(y– b1) =
(x– a1)(b2– b1).
EXPANDING BRACKETS
and rearrang-
ing terms again yields the general equation:
cx + dy = r
for some constants c, d, and r. This is called the general
form of the equation of a line. For instance, the equa-
tion of the vertical line three units to the right of the y-
axis is obtained by selecting c= 1, d= 0, and r= 3, to
yield the equation x= 3.
Some authors prefer to divide the general form of
the equation of a line through by the constant rand
change the names of the remaining labels so that the
equation reads:
+ = 1
This is called the intercept form of the equation of a line.
In three-dimensional space a line is specified by a
point (a,b,c) on the line and a
VECTOR
by v= <v1,v2,v3>,
representing the direction of the line. Thus the coordi-
nates (x,y,z) of any other point on the line are given by:
x= a+ tv1
y= b+ tv2
z= c+ tv3
for some value of the real number t. These are the
PARA
-
METRIC EQUATIONS
of the line. (The parametric equa-
tions of a line in two-dimensional space are analogous.)
If the vector vis computed via the difference of
coordinates of the point (a,b,c) and a second point
(a1,b1,c1) on the line—i.e., v1= a1– a,, v2= b1– b, and
v3= c1– c—then solving for tin the parametric equa-
tions yields:
= =
These are the “two-point form” equations of a line in
three-dimensional space.
See also
DIRECTION COSINES
;
LINEAR EQUATION
;
SIMULTANEOUS LINEAR EQUATIONS
;
SKEW LINES
;
VECTOR
EQUATION OF A PLANE
.
equation of a plane See
VECTOR EQUATION OF A
PLANE
.
equiangular A
POLYGON
is said to be equiangular if
all of its interior angles are equal. For example, a rect-
angle is equiangular (each interior angle equals 90°), as
is an
EQUILATERAL
triangle (each interior angle equals
60°). A polygon is called regular if it is both equiangu-
lar and equilateral.
A point (x,y) in the Cartesian plane is said to be a
lattice point if both xand yare integers, and a polygon
drawn in the plane is said to be a lattice polygon if its
z– c
–—
c1– c
y– b
–—
b1– b
x– a
–—
a1– a
x
—
b
x
—
a
y– b1
–—
x– a1
b2– b1
––—
a2– a1
2
–
–
3
5 – 3
–—
–1 – 2
y– 3
–—
x– 2
b2– b1
––—
a2– a1
y– b1
–—
x– a1
164 equation of a plane