Higher Engineering Mathematics, Sixth Edition

Vectors 263

From the geometry of the vector triangle,

the magnitudeofpq=

√

452 + 552 = 71 .06km/h

and the direction ofpq=tan−^1

(

55

45

)

= 50. 71 ◦

i.e.the velocity of carPrelative to carQis

71.06km/h at 50. 71 ◦

(a) (b) (c)

P Q

E

N

W

S

55 km/h

45 km/h

p

e

q

p e

q

Figure 24.40

(ii) The velocity of carQrelative to carPis given by
the vector equationqp=qe+epand the vector
diagram is as shown in Fig. 24.40(c), havingep
opposite in direction tope.
From thegeometry of this vector triangle,themag-
nitude ofqp=

√

452 + 552 = 71 .06m/s and the

direction ofqp=tan−^1

(

55

45

)

= 50. 71 ◦but must

lie in the third quadrant, i.e. the required angle is:

180 ◦+ 50. 71 ◦= 230. 71 ◦

i.e.the velocity of carQrelative to carPis

71.06m/s at 230. 71 ◦

Now try the following exercise

Exercise 105 Further problemson relative

velocity

1. A car is moving along a straight horizontal
   road at 79.2km/h and rain is falling vertically
   downwards at 26.4km/h. Find the velocity of
   the rain relative to the driver of the car.
   [83.5km/h at 71. 6 ◦to the vertical]


2. Calculate the time needed to swim across a
   river 142m wide when the swimmer can swim
   at 2km/h in still water and the river is flowing
   at 1 km/h. At what angle to the bank should
   the swimmer swim?
   [4minutes 55seconds, 60◦]
   3. A ship is heading in a direction N 60◦Eata
   speed which in still water would be 20km/h.
   It is carried off course by a current of 8km/h
   in a direction of E 50◦S. Calculate the ship’s
   actual speed and direction.
   [22.79km/h, E 9. 78 ◦N]

24.9 i,jandknotation

A method of completely specifying the direction of a

vector in space relative to some reference point is to use

three unit vectors,i,jandk, mutually at right angles

to each other, as shown in Fig. 24.41.

y

x

0

z

k

j

i

Figure 24.41

Calculations involving vectors given ini,jknotation

are carried out in exactly the same way as standard

algebraic calculations, as shown in the worked example

below.

Problem 14. Determine:

( 3 i+ 2 j+ 2 k)−( 4 i− 3 j+ 2 k)

( 3 i+ 2 j+ 2 k)−( 4 i− 3 j+ 2 k)= 3 i+ 2 j+ 2 k

− 4 i+ 3 j− 2 k

=−i+ 5 j

Problem 15. Givenp= 3 i+ 2 k,

q= 4 i− 2 j+ 3 kandr=− 3 i+ 5 j− 4 k

determine: