Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.7 EQUATIONS OF LINES, PLANES AND SPHERES

O

A

R

a

b

r

Figure 7.12 The equation of a line. The vectorbis in the directionARand

λbis the vector fromAtoR.

sinceRcan be reached by starting fromO, going along the translation vector

ato the pointAon the line and then adding some multipleλbof the vectorb.

Different values ofλgive different pointsRon the line.

Taking the components of (7.39), we see that the equation of the line can also

bewrittenintheform

x−ax

bx

=

y−ay

by

=

z−az

bz

= constant. (7.40)

Taking the vector product of (7.39) withband remembering thatb×b= 0 gives

an alternative equation for the line

(r−a)×b= 0.

We may also find the equation of the line that passes through two fixed points

AandCwith position vectorsaandc.SinceACis given byc−a, the position

vector of a general point on the line is

r=a+λ(c−a).

7.7.2 Equation of a plane

The equation of a plane through a pointAwith position vectoraand perpendic-

ular to a unit position vectornˆ(see figure 7.13) is

(r−a)·nˆ=0. (7.41)

This follows since the vector joiningAto a general pointRwith position vector

risr−a;rwill lie in the plane if this vector is perpendicular to the normal to

the plane. Rewriting (7.41) asr·nˆ=a·nˆ, we see that the equation of the plane

may also be expressed in the formr·nˆ=d, or in component form as

lx+my+nz=d, (7.42)