If the vector is the position vector of a point, then the lengths of the two components are called the coordinates of the point with reference to two lines, called the axes of coordinates, drawn through the origin of the vector.

Thus if P is any point, O an origin, then OP = p is the position vector.

Let the directions be given by the orts x and λ, then Ox and Oy drawn through O in these directions are the axes of coordinates.

Let a and be the components of p in the given directions, x and y their lengths, then x and y are the coordinates of P.

x is often called the abscissa and y the ordinate. We have at once the following relations.

It is seen therefore that given p and the axes of coordinates, x and y are known, and conversely given x and y, p is known.

Hence a vector in a plane involves two scalars.

47.

Theorem : If two vectors are equal, their components are equal.

Because we get for the determination of the components, two triangles like OAP equal in all respects.

48. If the two axes are perpendicular, the coordinates are called rectangular and the orts are denoted by 1 and 2, so that

By means of relations similar to the above, we can always express any vector equation (coplanar vectors) as two scalar equations, for instance if

then by expressing p, a and B in terms of their components, we obtain two equations between coordinates, say

Equation (i) gives us the same information as equations

(ii) and (iii) conjointly.

Let r denote the length of p, then

p2 = x2 + y2.

If the angle POx be denoted by 0, then

x = r cos 0, y = r sin 0

Hence if we know the rectangular components of p we oan calculate the magnitude of the vector and determine its direction and sense.

It should be noticed that the axes of x and y (extending in both the positive and negative senses) divide the plane into four quadrants and that each of these quadrants has its own combination of coordinate signs. Knowing then the signs of the coordinates x and y we know at once in which quadrant the position vector is situated. For instance, if x is positive and y is negative, then p must lie in the 4th quadrant.

y

49. If a and ẞ be the position vectors of two points A and B, then AB as a vector is given by B-a, which involves the magnitude, direction and sense of AB. Let (a, b) and (c, d) be the rectangular coordinates of A and B, then

B

(1) The position vectors of three points are given by

respectively, find as vectors the sides of the triangle having the given points as vertices.

(2) The rectangular coordinates of three points are (2, 3), (1, 0), (−2, 1), find as vectors the lines joining these points.

(3) Find the magnitude of the position vectors + 32 and 611-22; find also the magnitude of the vectors joining the given points, and hence shew that the given vectors are at right angles.

(4) Determine the lengths of the lines joining the four points taken in order whose position vectors are

311 + 212, 411 + 42, 242 + 541 and 311⁄2 + 44

and shew that they form a rectangle.

(5) The coordinates of two points are

Find

x = 2, y = 3; x = −

- 1, y = 4.

(1) the vector joining the points in terms of the direction orts, (2) the length of the line), the axes being at right angles. (3) the direction

DECOMPOSITION OF VECTORS IN SPACE.

50. We saw (§42) that between three vectors not in a plane no relation is possible, and such vectors were therefore said to be independent.

Let a, B, y, be three independent vectors, and O their common initial point.

In the figure, ẞ and y may be supposed in the plane of the paper, and a to stick upwards towards the reader.

To perform the summation a+B+y, we may move B parallel to itself to the end point A of a and thus get

OS = a + B.

The figure OASB is obviously by construction a parallelogram.

Now suppose OASB is moved parallel to itself so that the point O moves along y to C.

Then the points A, B, and S will move along y to R, Q, and P respectively, and the parallelogram OASB will sweep out a volume, which volume is called a parallelopiped.

The vector OP = a + B + y.

From the method of generating the parallelopiped we see at once that

From the figure it is evident that, in whatever order we add a, ẞ and y, we always come to the point P.

The line OP is called a diagonal of the parallelopiped; it is a line joining opposite corners. There are three other such lines, viz. AQ, SC and BR.

Calling OP p, we have

p=a+B+y.

Hence given three vectors in space, we can form their sum; it is the diagonal of the parallelopiped of which the