The elements of coordinate geometry, by S. L. Loney, M.A.

[Exs. II.]

POLAR COORDINATES.

19

Find the areas of the quadrilaterals the coordinates of whose
angular points, taken in order, are
13. (1, 1), (3, 4), (5, -2), and (4, -7).
14. (-1, 6), (-3, -9), (5, -8), and (3, 9).
15. If O be the origin, and if the coordinates of any two points
P1 and P2 be respectively (xi, y) and (x2, Y2), prove that
OP. OP2. cos P1OP2 = x1X2 + Y1Y.
30. Polar Coordinates. There is another method,
which is often used, for determining the position of a point
in a plane.
Suppose 0 to be a fixed point, called the origin or
pole, and OX a fixed line, called the initial line.
Take any other point P in the plane of the paper and
join OP. The position of P is clearly known when the
angle XOP and the length OP are given.
[For giving the angle XOP shews the direction in which OP is
drawn, and giving the distance OP tells the distance of P along this
direction.]
The angle XOP which would be traced out by the line
OP in revolving from the initial line OX is called the
vectorial angle of P and the length OP is called its radius
vector. The two taken together are called the polar coordinates of P.
If the vectorial angle be 0 and the radius vector be r, the
position of P is denoted by the symbol (r, 0).
The radius vector is positive if it be measured from the
origin 0 along the line bounding the vectorial angle; if
measured in the opposite direction it is negative.
31. Ex. Construct the positions of the points (i) (2, 30~),
(ii) (3, 150~), (iii) (-2, 45~), (iv)
(-3, 330~), (v) (3, -210~) and (vi). L
(-3, -30~). \
(i) To construct the first point,
let the radius vector revolve from --
OX through an angle of 30~, and
then mark off along it a distance
equal to two units of length. We p M
thus obtain the point P1.
(ii) For the second point, the radius vector revolves from OX
through 150~ and is then in the position OP2; measuring a distance 3
along it we arrive at P2.
2-2