If u denotes any physical unit, n a number, an ort, then a vector quantity can be expressed in the form

пиш.

So that multiplying a vector quantity by a number or scalar means multiplying its magnitude by that number or scalar.

37. Subtraction of Vectors. We have seen (§ 19) that any step AB - the step BA, because the two steps AB+ BA bring us back to the point from which we started. Hence multiplying a vector by -1, means change its sense, so that if AB = a, and BA = a', then a and a + a' = 0.

According to the definition of subtraction in § 17 we understand by the difference of two vectors a- ß, that vector y which added to ẞ gives a or

Set off from the same origin O the vectors a and ß,

α

OA = a

in figure
OBB, we get

OB + BA=0A

BAOA OB,

A

or

and

Fig. 18.

structed in a different

manner;

point of a set off AB'B' - ẞ, then

B can be con

Hence the rule:-In order to subtract a vector, change its sense and add.

38. Theorem: In a sum of Vectors, if each vector be multiplied by the same number (or scalar), the sum is multiplied by that number (or scalar).

The order of the addition of Vectors was shewn in § 32 to be immaterial; hence if we have n equal vectors a and n equal vectors B, we may form the sum

(i) by adding to the sum of all the a's the sum of all the B's, this gives

na + nẞ,

(ii) by adding the a's and B's in pairs, this gives n (a + B).

Hence

na + nẞ = n (a + B),

where n is any whole number.

This can be extended to include any number, fractional, incommensurable, negative.

Geometrically the theorem is that of Euc. VI. vi., from which the general case follows at once.

If two triangles have one angle of the one equal to one angle of the other and the sides about the equal angles proportionals, the triangles shall be similar."

For if one triangle be applied to the other so that the

proportional sides are in the same straight lines, then if a and B denote the two sides of the one, na and nẞ will denote the corresponding sides of the other.

The third sides will then (see fig.) be a +B and na+nẞ; but the

theorem says that the third sides must be parallel and proportional and hence na + nẞ must be the same as n (a + B).

39. It has now been shewn that for vector addition all the rules of ordinary Algebra hold, sense taking the place of sign.

40. Between any two parallel vectors there is a relation, viz. one is a multiple of the other; no such relation is possible between two non-parallel vectors and any two non-parallel vectors are said to be independent. If then we have such an equation as

aa+bB = = ca + dB

and we know that a and B are not parallel, then
a = c and b = d.

For from the above equation we have at once
(a — c) a = (d — b) B

and since a is not a multiple of ẞ the equation can only be true if both sides vanish.

41. Theorem: Between any three non-parallel coplanar vectors there is a relation, viz. any one of them can be expressed in terms of the other two.

This construction can be done in two ways as seen from the figure, but both give the same components aa and bẞ. No other triangle can be drawn having OB as one side and the other sides parallel to a and B.

Since any multiple of y can be expressed in a similar way, we may say that in general between any three coplanar vectors there is a relation of the form

la + mB + ny = 0.

Conversely if three vectors a, B, y be connected by an equation of the form

αα

cr
Fig. 21.

aa+bB+cy = 0,

then a, B, y are coplanar.

For the equation says that multiples of the three vectors are the three sides of a triangle.

42. If three vectors are not in a plane, no such relation is possible and the three vectors are independent.

43. In § 41 we saw that if three vectors a, B, y are connected by a relation of the form

then the end points of the three vectors when drawn from a common origin are collinear.

Let a and B be the position vectors of two points A and B with reference to some origin O, and let Ĉ be any point in AB or AB produced and y its position vector.

in which it is seen that the sum of the coefficients of the vectors is zero.

44.

A very important special case of this is when C

is the mid-point of AB.

This could also be derived directly from the general case by putting t = − 1.

SECTION (II). APPLICATIONS OF VECTORS
TO GEOMETRY.

45. In order that the student may gain facility in working and thinking in vectors, we will give some of the applications of vectors to geometry. Many geometrical theorems can be established with great ease by the aid of vectors.

(i) If the diagonals of a quadrilateral bisect one another, the figure is a parallelogram.

Let AB and CD be the bisecting diagonals and the point of bisection.