Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR ALGEBRA

O

a d

ˆn

r

AR

Figure 7.13 The equation of the plane is (r−a)·ˆn=0.

where the unit normal to the plane isˆn=li+mj+nkandd=a·nˆis the

perpendicular distance of the plane from the origin.

The equation of a plane containing pointsa,bandcis

r=a+λ(b−a)+μ(c−a).

This is apparent because starting from the pointain the plane, all other points

may be reached by moving a distance along each of two (non-parallel) directions

in the plane. Two such directions are given byb−aandc−a. It can be shown

that the equation of this plane may also be written in the more symmetrical form

r=αa+βb+γc,

whereα+β+γ=1.

Find the direction of the line of intersection of the two planesx+3y−z=5and

2 x− 2 y+4z=3.

The two planes have normal vectorsn 1 =i+3j−kandn 2 =2i− 2 j+4k.Itisclear
that these are not parallel vectors and so the planes must intersect along some line. The
directionpof this line must be parallel to both planes and hence perpendicular to both
normals. Therefore

p=n 1 ×n 2

= [(3)(4)−(−2)(−1)]i+[(−1)(2)−(1)(4)]j+ [(1)(−2)−(3)(2)]k

=10i− 6 j− 8 k.

7.7.3 Equation of a sphere

Clearly, the defining property of a sphere is that all points on it are equidistant

from a fixed point in space and that the common distance is equal to the radius