Vectors as Extension of Numbers

   The idea of numbers developed very gradually. First came positive integers-- 1,2,3... (not zero, a much more recent addition) to describe countable objects, such as sheep, days, tribe members, etc.

   The concept of negative numbers might have arisen as an extension of subtraction, or maybe from money--what you owed was negative wealth, red ink in the ledger.

   Objects which could be divided--for instance, land--brought fractions. Then around 500 BC a student of Pythagoras proved that the number giving the square root of 2 could never be written as a fraction; it did not make sense to them, and therefore we still say these are "irrational" numbers. Between them, integers, fractions and irrationals can describe anything that has a size, a magnitude.

But what can describe velocity, which has both a magnitude and a direction?

A vector, that's what.

Velocities can be added. Suppose an airplane flies at 200 mph (or if you wish, kilometers per hour) with a tail wind of 100 mph. How fast does it cover distance on the ground? Easy: for every 200 miles it advances, the wind carries it 100 miles further, so the answer is

Graphically, each stretch on the ground or velocity can be represented by an arrow in its direction, its length giving the magnitude--for instance, an arrow AB 200 mm (milimeters) long for the airplane's motion and another one BC 100 mm long, in the same direction, for the wind. To add velocities, put them end to end, as in the top figure here.

and the "arrow addition" confirms it (middle figure, lengths distorted).

Next suppose the pilot's course is eastwards, but a 100 mph cross-wind blows towards the north-east: in what direction does the airplane move, and how fast? Intuition does not help, but arrow addition does (bottom figure, not to scale).

The general rule is, the combined velocities bring the airplane in an hour to the same spot as when first one motion, then the other, acts alone for an hour. As expected, the direction is somewhere between eastward and northward.

  All vectors can be added together in this way--like arrows, head to tail. There exists an alternate method, though, often easier to use, and it is described next.

Resolving vectors into components

  Suppose the given single vector is represented by the arrow AB in the drawing, and we wish to resolve it into the sum of two vectors directed along AA' and AA". We draw lines along AA' and AA" and also lines parallel to them from B, the other end of the vector. If AA' and AA" are perpendicular to each other (the usual case), then these lines enclose a rectangle ACBD, of which AB is a diagonal. It is now evident that AC and CB are a solution to our problem, and in vector addition

AC + CB = AB

The Uses of Component Vectors

(a) Adding many vectors

  A faster way is to choose two perpendicular directions, and taking our cue from cartesian coordinates, one will be named here "the x direction" and the other "the y direction". Resolve each vector V into an "x component" V_x along the x direction and a "y component" V_y along the y direction.

   Now we have not 10 but 20 vectors which need to be added up, but the job is much easier. Of those vectors, 10 are lined up with the x-direction, and vectors in the same direction (like head- and tail-wind velocities in the earlier example of the airplane) add up like ordinary numbers. The same goes for the 10 vectors lined up with the y-direction. The problem is now reduced to two strings of ordinary additionas and subtractions (oppositely directed vectors have minus sign), and only the last step--adding together the grand totals in the x and y directions--requires vector-type addition.

Note: The real world is three dimensional, and so are its vectors. But 3-D vectors can also be resolved, each of them equal to the sum of three vectors in the (x, y, z) directions (the rectangle now becomes a rectangular box). Components of the same kind are added as in the 2-dimensional example above, and the final addition just involves one sum-vector in each dimension. The vector sum is now the diagonal of the rectangular box of which the three sums are the sides.

(B) Calculating a vector sum

[The values of (A,B) can change if the same vector is represented in different coordinates, but that is just its representation mode. Similarly, a temperature is the same if represented in degrees centigrade or farenheit, although the numbers representing it may differ.]

Let the x-direction be eastward and the y-direction northward. Then the (x,y) velocity components are, in mph,

• --of the airspeed, (200,0)
• --of the wind velocity (100 cos 45^o,100 sin 45^o) = (70.7, 70.7)

This gives the total velocity V. By the theorem of Pythagoras,

giving the magnitude of V as approximately 280 mph, while the sharp angle at point A in the figure (we will name it A, too) satisfies

From which A is about 16.25 degrees.

(c) The Inclined Plane

• One is perpendicular to the surface, is represented by the line AC and has magnitude W cos s. That force is completely cancelled by the resistance of the surface, which allows no motion in that direction. If the motion includes friction, however, the force of friction is proportional to this component.

• The other force is parallel to the surface, is represented by the line AD, has magnitude W sin s and, if no friction impedes the motion, is free to accelerate the block in that direction. The force is smaller that the weight W by a factor (multiplier) sin s, a number always smaller than 1, while the mass of the block is unchanged. Its acceleration is therefore reduced by a similar factor and does not equal g as it would for free fall, but only g sin s.

Further Exploring