Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.8 USING VECTORS TO FIND DISTANCES

O

P

d

a

p

nˆ

Figure 7.15 The minimum distancedfrom a point to a plane.

Find the distance from the pointPwith coordinates(1, 2 ,3)to the plane that contains the

pointsA,BandChaving coordinates(0, 1 ,0),(2, 3 ,1)and(5, 7 ,2).

Let us denote the position vectors of the pointsA, B, Cbya,b,c. Two vectors in the
plane are

b−a=2i+2j+k and c−a=5i+6j+2k,

and hence a vector normal to the plane is

n=(2i+2j+k)×(5i+6j+2k)=− 2 i+j+2k,

and its unit normal is

ˆn=

n

|n|

=^13 (− 2 i+j+2k).

Denoting the position vector ofPbyp, the minimum distance from the plane toPis
given by

d=(a−p)·nˆ

=(−i−j− 3 k)·^13 (− 2 i+j+2k)

=^23 −^13 −2=−^53.

If we takePto be the originO, then we findd=^13 , i.e. a positive quantity. It follows from
this that the original pointPwith coordinates (1, 2 ,3), for whichdwas negative, is on the
opposite side of the plane from the origin.

7.8.3 Distance from a line to a line

Consider two lines in the directionsaandb, as shown in figure 7.16. Sincea×b

is by definition perpendicular to bothaandb, the unit vector normal to both

these lines is

nˆ=

a×b

|a×b|

.