Basic Engineering Mathematics, Fifth Edition

Vectors 277

From the geometry of thisvector triangle, the mag-

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 ◦

That is,the velocity of carQrelative to carPis

71.06m/s at 230. 71 ◦

Now try the following Practice Exercise

PracticeExercise 116 Relative velocity

(answers on page 352)

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.


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 1km/h. At what angle to the bank should
   the swimmer swim?


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.

29.9 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 Figure 29.41.

y

x

z

k

i o j

Figure 29.41

Calculations involving vectors given ini,j,knota-

tion are carried out in exactly the same way as standard

algebraic calculations, as shown in the workedexamples

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

(a)−r(b) 3p(c) 2p+ 3 q(d)−p+ 2 r

(e) 0. 2 p+ 0. 6 q− 3. 2 r

(a) −r=−(− 3 i+ 5 j− 4 k)=+ 3 i− 5 j+ 4 k

(b) 3p= 3 ( 3 i+ 2 k)= 9 i+ 6 k

(c) 2p+ 3 q= 2 ( 3 i+ 2 k)+ 3 ( 4 i− 2 j+ 3 k)

= 6 i+ 4 k+ 12 i− 6 j+ 9 k

= 18 i− 6 j+ 13 k

(d) −p+ 2 r=−( 3 i+ 2 k)+ 2 (− 3 i+ 5 j− 4 k)

=− 3 i− 2 k+(− 6 i+ 10 j− 8 k)

=− 3 i− 2 k− 6 i+ 10 j− 8 k

=− 9 i+ 10 j− 10 k

(e) 0. 2 p+ 0. 6 q− 3. 2 r

= 0. 2 ( 3 i+ 2 k)+ 0. 6 ( 4 i− 2 j+ 3 k)

− 3. 2 (− 3 i+ 5 j− 4 k)

= 0. 6 i+ 0. 4 k+ 2. 4 i− 1. 2 j+ 1. 8 k+ 9. 6 i

− 16 j+ 12. 8 k

= 12. 6 i− 17. 2 j+ 15 k

Now try the following Practice Exercise

PracticeExercise 117 i, j, knotation

(answers on page 352)

Given thatp= 2 i+ 0. 5 j− 3 k,q=−i+j+ 4 k

andr= 6 j− 5 k, evaluateand simplifythefollow-

ing vectors ini,j,kform.

1. −q 2. 2p


2. q+r 4. −q+ 2 p


3. 3q+ 4 r 6. q− 2 p


4. p+q+r 8. p+ 2 q+ 3 r


5. 2p+ 0. 4 q+ 0. 5 r 10. 7r− 2 q