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 m of the line.
Thus one can rewrite the two-point form of the equation
as
. Rearranging yields:
y – b 1 = m(x – a 1 )
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
+ (ma 1 + b 1 ). Denoting the constant ma 1 + b 1 simply as
b yields the equation:
y = mx + b
Noting that if x = 0, then we have y = b. This shows
that the constant b is the y-intercept of the line. For
this reason, the above equation is called the “slopeintercept form” equation of the line. Thus, for example, 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 (a 2 – a 1 )(y – b 1 ) =
(x – a 1 )(b 2 – b 1 ). EXPANDING BRACKETS and rearranging 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 equation of the vertical line three units to the right of the yaxis 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 r and
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 =,
representing the direction of the line. Thus the coordinates (x,y,z) of any other point on the line are given by:
x = a + tv 1
y = b + tv 2
z = c + tv 3
for some value of the real number t. These are the PARAMETRIC EQUATIONS of the line. (The parametric equations of a line in two-dimensional space are analogous.)
If the vector v is computed via the difference of
coordinates of the point (a,b,c) and a second point
(a 1 ,b 1 ,c 1 ) on the line—i.e., v 1 = a 1 – a,, v 2 = b 1 – b, and
v 3 = c 1 – c—then solving for t in the parametric equations 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 rectangle 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 equiangular and equilateral.
A point (x,y) in the Cartesian plane is said to be a
lattice point if both x and y are integers, and a polygon
drawn in the plane is said to be a lattice polygon if its
z – c
–—
c 1 – c
y – b
–—
b 1 – b
x – a
–— a 1 – a
x
— b
x
— a
y – b 1
–— x – a 1
b 2 – b 1
––— a 2 – a 1
2
– –
3
5 – 3
–—
–1 – 2
y – 3
–—
x – 2
b 2 – b 1
––— a 2 – a 1
y – b 1
–— x – a 1
164 equation of a plane
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 m of the line.
Thus one can rewrite the two-point form of the equation
as
. Rearranging yields:
y – b 1 = m(x – a 1 )
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
+ (ma 1 + b 1 ). Denoting the constant ma 1 + b 1 simply as
b yields the equation:
y = mx + b
Noting that if x = 0, then we have y = b. This shows
that the constant b is the y-intercept of the line. For
this reason, the above equation is called the “slopeintercept form” equation of the line. Thus, for example, 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 (a 2 – a 1 )(y – b 1 ) =
(x – a 1 )(b 2 – b 1 ). EXPANDING BRACKETS and rearranging 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 equation of the vertical line three units to the right of the yaxis 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 r and
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 =
representing the direction of the line. Thus the coordinates (x,y,z) of any other point on the line are given by:
x = a + tv 1
y = b + tv 2
z = c + tv 3
for some value of the real number t. These are the PARAMETRIC EQUATIONS of the line. (The parametric equations of a line in two-dimensional space are analogous.)
If the vector v is computed via the difference of
coordinates of the point (a,b,c) and a second point
(a 1 ,b 1 ,c 1 ) on the line—i.e., v 1 = a 1 – a,, v 2 = b 1 – b, and
v 3 = c 1 – c—then solving for t in the parametric equations 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 rectangle 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 equiangular and equilateral.
A point (x,y) in the Cartesian plane is said to be a
lattice point if both x and y are integers, and a polygon
drawn in the plane is said to be a lattice polygon if its
z – c
–—
c 1 – c
y – b
–—
b 1 – b
x – a
–— a 1 – a
x
— b
x
— a
y – b 1
–— x – a 1
b 2 – b 1
––— a 2 – a 1
2
– –
3
5 – 3
–—
–1 – 2
y – 3
–—
x – 2
b 2 – b 1
––— a 2 – a 1
y – b 1
–— x – a 1
164 equation of a plane