Calculus: Analytic Geometry and Calculus, with Vectors

1.3 Lines and linear equations 17

which employs the standard trigonometric for-
mula for the tangent of the difference of two

angles. When we are asked to find 0, we pre-
sent tan c as the answert to our problem.
As an application of some of the above ideas,
we find the equation of the line L of Figure

1.38. The positive number p is the distance

Figure 1.38

from the origin to the point P1, and L is perpendicular to the line
OP1 at Pl. The coordinates of Pi are p cos a (alpha) and p sin a. The
slope of OPl is tan a, or sin a/cos a, and the slope of L is the negative
reciprocal - cos a/sin a. Use of the point-slope formula gives the equa-
tion of L in the form

(1.381) y - p sin a = -

cos a (x

• p cos a).
  sin a

Multiplying by sin a and using the identity sine a + cos' a = 1 puts the
equation in the more attractive form

(1.382) (cos a)x + (sin a)y = p.

The line OPt, being a line perpendicular to L, is called a normal to L, and
the equation (1.382) is called the normal form of the equation of L because
it gives information about this normal. It is sometimes thought to be
worthwhile to know a speedy way to put the equation 14x + By + C = 0
into normal form. We suppose that 14 and B are not both 0 and that

C s 0. The trick is to transpose C to obtain A4x + By = -C and then
divide by one of ± .s /A2 + B2 to obtain

14 B -C

(1.383) X + y =

± x/,12 + B2 -{- \/22 + B2 ± A2 + B2'

where the sign is so chosen that the right side is positive. This equation
has the normal form. The right side is the distance p from the origin to
the line, and the coefficients of x and y are respectively the numbers cos a
and sin a which determine the angle a of Figure 1.38.

Problems 1.39

1 Each of the following equations is the equation of a line L. In each case,

find the slope m by finding the coordinates of the points in which the line inter-

sects the coordinate axes and then finding the slope of the line through those

f Traditionally, students are required by tests and examinations to find tan 0. Accord-
ingly, students who hope to pass examinations by learning a few formulas-and those who
aspire to a substantial understanding of their subject-are well advised to learn the neces-
sary ritual.