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76. Distance has direction.

We have hitherto supposed all our distances to be measured in the same direction, (or in exactly opposite directions), so that it was only necessary to consider the magnitude and sign of our distances.

In what follows we shall always consider the direction of a distance to be an essential part of it. As a necessary consequence we must consider direction to be an essential part also of all quantities which vary directly as distance, such as velocity, acceleration, force, impulse.

77. Quantities of which direction is an essential part, must be carefully distinguished from those quantities in which it is not.

Quantities having direction are called vector quantities. Distance, velocity, acceleration, force, impulse have direction.

Quantities without direction are called scalar quantities. Time, mass, volume, speed have no direction.

78. It is convenient to use the word speed to denote the magnitude only of a velocity ; just as we use the word length to denote the magnitude only of a distance. The average speed of a point moving in a curved line is that which varies directly as the length of the curved line passed over by the point, and inversely as the interval occupied.

The average speed of a point moving in a straight line is the same as its average velocity.

When we speak of a train moving at the rate of 40 miles an hour for a certain interval, we mean that its average speed is 40 miles per hour.

A point may have a constant speed while its velocity is changing (in direction).

Similarly we may use the word quickening to denote the magnitude only of an acceleration.

79. PROP. Any quantity which has direction and magnitude may be represented by a finite straight line with an arrow-head.

For the direction of the quantity can be represented by the di tion of the line with the arrow-head, and the numerical measure of the quantity can be indicated by that of the length of the line.

When a distance is indicated by the two letters at its extremities, the direction of the arrow-head may be indicated by the order of the letters.

80. A distance can always be found which represents the combined effect, or the resultant, of two given distances in different directions.

The combined effect of, (or the result of adding together) two nondirectional, (or scalar) quantities is the arithmetical sum of their measures.

The same is true of two directional, (or vector) quantities which have the same direction.

But the combined effect of two directional, (or vector) quantities having different directions requires definition.

Example. A point P is said to have a distance of 2 inches northwards from a given point O, and also a distance of 3 in. towards the

і north-west from O, when the position of Pis found as follows:

Starting from O measure OQ 2 in. towards the north ; then from @ measure QP 2 in. towards the north-west.

Or thus; starting from O measure OQ zin. towards the north-west and then from l' measure QP 2 in. towards the north.

It will be seen that the resulting position of P is the same in each case; for OQ, og are sides of a parallelogram and P is at the corner opposite to 0.

81. We may illustrate the meaning of a point having two simultaneous distances from a starting point thus :

Suppose a sheet of glass to be placed on paper and the point P to be a moveable mark placed on the glass.

Let the mark be moved a distance of 3in. on the glass towards the north-west, and let the glass be moved relatively to the paper, a distance of 2 in. northwards.

It will be seen that the resulting position of the point P with reference to the paper is that found as in the Example in Art. 80.

And it makes no difference to the ultimate position of P whether these distances are passed over simultaneously, or whether they are passed over separately, one of the movements being finished before the other is made.

Example i. A point A is distant 3 miles towards the North and 4 miles towards the West from a point B; what is the total distance of A from B?


Draw BC northwards 3 units of length ; draw BD westwards 4 units of length; complete the parallelogram CBDA ; then CBDA is a right-angled parallelogram and

AB2= ADP + BD2. Hence if AB contains x units of length

x=32 +.42=9+16=25; therefore

Thus the distance AB is 5 units of length.

Hence it follows that the total distance of the two places described in the question from each other is 5

miles. And the direction is such that if AB make an angle a with BD,

DA then, tan a=


Example ii. A man walks 20 yards eastwards and then 25 yards in a direction making an angle whose sine is it with the line pointing eastwards and on the north side of it; find the direction and magnitude of the resultant distance of the man from his starting point.

Let OQ represent a distance of 20 yards eastwards; produce OQ to N; let QP represent the distance of 25 yds., QP being drawn so that sin PQN=

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Draw PN perpendicular to OQ produced; then, NP

is the sine of the angle which QP makes with ON; therefore

Let PN contain y yds. Then y=x of 25 = 15.
Similarly, QN may be shewn to contain 20 yds.
Let OP contain x yds. Then since OP=ON2 + NP?


=1825; therefore,

=(1825) = 42°7... Hence, the required distance is 42°7 yds., also its direction makes an angle whose tangent is 16, or $, with the line towards the East.

EXAMPLES. XXIII. 1. A man walks 300 yards northwards and then 400 yards westwards; another man starting from the same place walks 400 yards westwards and then 300 yards northwards; shew that the two men are then at the same total distance from the starting point; and find that distance.

2. A dog on an ice-floe goes 3 miles northwards while the floe itself travels 4 miles eastwards; how far is the dog then from the place on the earth's surface from which he started ?

3. A fly crawls along the floor of a railway carriage a distance of 2 ft., at right angles to the line of rails, and at the same time the carriage moves a distance of 10 ft.; find approximately the distance moved over by the fly.

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4. A man swims across a river, and in 40 seconds he swims 20 yards; in the same time the current of the river has carried him 15 yards down stream ; how far is he from his starting point after

40 secs. ?

5. A fly, in a railway carriage which is moving at the rate of 10 ft. per sec. northwards, crawls at the rate of 5 ft. in 10 seconds across the Aoor of the carriage towards the north-east; what distance will the fly have gone in 10 seconds?

6. On the floor of a railway carriage which is moving with 4 velos northwards, four flies A, B, C, D start from the same place and crawl each with 3 velos, A Northwards, B Southwards, C Westwards, and D

3 towards the North-West. Find their distances from the starting point after 2 seconds.

The following examples require the use of the Trigonometric Ratios.

7. A man walks 350 yards in a certain direction and then turning his direction through an angle of 120 degrees walks 350 yards more; find the direction and magnitude of his resulting distance from his starting point.

8. A man on the deck of a ship moving due north, walks 20 yards towards the north-east while the ship goes 100 yds.; find the direction and magnitude of the whole distance passed over by the man.

9. The distance of a point B from another A is the sum of two equal distances of 20 yds. inclined to each other at an angle of 60 degrees; find the distance of B from A.

10. When the wind is due North a certain boat can sail either towards the north-east or north-west; it sails first 2 miles towards the N.E., then 2 miles towards the N.W., then i mile more towards the N.E. ; find the magnitude and direction of its resulting distance from its starting point.

11. I walk 3 miles northwards, then 2 miles in a direction pointing 30° to the east of north; what will then be my distance from my starting point ?

12. I walk 3 miles in a straight line, I then change my direction through an angle whose sine is and walk 5 miles straight on, I then turn (in the same direction as before) through a right angle and walk I mile; how far shall I then be from the place from which I started ?

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