Higher Engineering Mathematics, Sixth Edition

284 Higher Engineering Mathematics

magnitude, direction and sense), thenAP=λb,whereλ

is a scalar quantity. Hence, from above,

r=a+λb (8)

If, say,r=xi+yj+zk,a=a 1 i+a 2 j+a 3 kand

b=b 1 i+b 2 j+b 3 k, then from equation (8),

xi+yj+zk=(a 1 i+a 2 j+a 3 k)

+λ(b 1 i+b 2 j+b 3 k)

Hence x=a 1 +λb 1 , y=a 2 +λb 2 and z=a 3 +λb 3.

Solving forλgives:

x−a 1

b 1

=

y−a 2

b 2

=

z−a 3

b 3

=λ (9)

Equation(9)isthestandardCartesianformforthevector

equation of a straight line.

Problem 11. (a) Determine the vector equation of

the line through the point with position vector

2 i+ 3 j−kwhich is parallel to the vectori− 2 j+ 3 k.

(b) Find the point on the line corresponding toλ= 3

in the resulting equation of part (a).

(c) Express the vector equation of the line in

standard Cartesian form.

(a) From equation (8),

r=a+λb

i.e. r=( 2 i+ 3 j−k)+λ(i− 2 j+ 3 k)

or r=( 2 +λ)i+( 3 − 2 λ)j+( 3 λ− 1 )k

which is the vector equation of the line.

(b) Whenλ=3, r= 5 i− 3 j+ 8 k.

(c) From equation (9),

x−a 1

b 1

=

y−a 2

b 2

=

z−a 3

b 3

=λ

Since a= 2 i+ 3 j−k,thena 1 = 2 ,

a 2 =3anda 3 =−1and

b=i− 2 j+ 3 k,then

b 1 = 1 ,b 2 =−2andb 3 = 3

Hence, the Cartesian equations are:

x− 2

1

=

y− 3

− 2

=

z−(− 1 )

3

=λ

i.e. x− 2 =

3 −y

2

=

z+ 1

3

=λ

Problem 12. The equation

2 x− 1

3

=

y+ 4

3

=

−z+ 5

2

represents a straight line. Express this in vector

form.

Comparing the given equation with equation (9), shows

that the coefficients ofx,yandzneed to be equal to

unity.

Thus

2 x− 1

3

=

y+ 4

3

=

−z+ 5

2

becomes:

x−^12

3

2

=

y+ 4

3

=

z− 5

− 2

Again, comparing with equation (9), shows that

a 1 =

1

2

,a 2 =−4anda 3 =5and

b 1 =

3

2

,b 2 =3andb 3 =− 2

In vector form the equation is:

r=(a 1 +λb 1 )i+(a 2 +λb 2 )j+(a 3 +λb 3 )k,

from equation (8)

i.e.r=

(

1

2

+

3

2

λ

)

i+(− 4 + 3 λ)j+( 5 − 2 λ)k

orr=

1

2

( 1 + 3 λ)i+( 3 λ− 4 )j+( 5 − 2 λ)k

Now try the following exercise

Exercise 114 Further problems on the

vector equation of a line

1. Find the vector equation of the line through the
   point with position vector 5i− 2 j+ 3 kwhich