Cambridge Additional Mathematics

DEMO

A fixed point()

R any point()

O origin() line

b

a

r

296 Vectors (Chapter 11)

5 An aeroplane needs to fly due east from one city to another at a speed of 400 km h¡^1. However, a

50 km h¡^1 wind blows constantly from the north-east.

a How does the wind affect the speed of the aeroplane?

b In what direction must the aeroplane head to compensate for the wind?

We have seen in Cartesian geometry that we can determine theequation of a lineusing itsdirectionand

anyfixed pointon the line. We can do the same using vectors.

Suppose a line passes through a fixed point A with position

vectora, and that the line is parallel to the vectorb.

Consider a point R on the line so that

¡!

OR=r.

By vector addition,

¡!

OR=

¡!

OA+

¡!

AR

) r=a+

¡!

AR.

Since

¡!

AR is parallel tob,

¡!

AR=tb for some scalar t 2 R

) r=a+tb

Suppose a line passes through a fixed point A(a 1 ,a 2 ) with position vectora, and that the line is parallel

to the vector b=

μ

b 1

b 2

¶

.IfR(x,y) with position vectorris any point on the line, then:

² r=a+tb, t 2 R or

μ

x

y

¶

=

μ

a 1

a 2

¶

+t

μ

b 1

b 2

¶

is thevector equationof the line.

² The gradient of the line is m=

b 2

b 1

² Since

μ

x

y

¶

=

μ

a 1 +b 1 t

a 2 +b 2 t

¶

, theparametric equationsof

the line are x=a 1 +b 1 t and y=a 2 +b 2 t, where t 2 R

is theparameter.

Each point on the line corresponds to exactly one value oft.

² We can convert these equations into

Cartesian form by equatingtvalues.

Using t=

x¡a 1

b 1

=

y¡a 2

b 2

we obtain

b 2 x¡b 1 y=b 2 a 1 ¡b 1 a 2 which is the

Cartesian equationof the line.

G Lines

R,(x y)

A,(a a ) 12

= b ______

b ______

1

2

__

b {}_

It is possible to convert

between vectors and

Cartesian equations.

However, in and higher

dimensions, vectors are

much simpler to use.

3

The equations of

lines do not need

to be written in

parametric form

for the syllabus.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_11\296CamAdd_11.cdr Friday, 4 April 2014 2:25:17 PM BRIAN