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52 4] COORDINATE GEOMETRY IN A PLANE This may be rewritten in column vector notation (with i, j as base) Y = (x) a (x2— x1) + Y2 Y1 • By the Uniqueness Theorem in two dimensions this gives the equations (x x1= A(x2 — x1), — { (Y —Yi = A(Y2 —Y1). Eliminating a between these two equations we obtain the following result. The Cartesian equation of the line joining the points P1(x1, Yi)P2(x2,Y2) is x—x1= y— yl x2 —x1 Y2—Y1 (The reader must note carefully the distinction in this equation between x, y on the one hand and x1, yl, x2, y2on the other: x, y are the coordinates of a general point P on the line; x1, yi, x2, Y2 are the coordinates of two specified fixed points of the line.

Deduce the position vector of X, the intersection of BM and CN. 8. ABCDA'B'C'D' is a parallelepiped, with parallel edges AA', BB', CC', DD'. U is the point of trisection of AA' nearer A, V is the point of trisection of C'D' nearer C' and W is the mid-point of B'C'. With A as origin, the position vectors of B, D, A' are respectively b, d, a'. Write down the position vectors of U, V, W and a general point of DD'. Locate the point at which the plane UVW cuts DD'. 9. OUVW is a tetrahedron, the position vectors of U, V, W being u, v, w.

19. Find the equations of the lines joining the pairs of points: (ii) (0, 1), (2, —1); (i) (2, — 3), (3, 2); (iii) (-3, —1), (-1, 2); (iv) (1, 2), (-3, 2). Ex. 20. What are the gradients of the following lines: (i) 2x—y-3 = 0; (ii) x+y+1 = 0; (iii) 3x+4y+2 = 0? Suppose now we have two lines, L1and L2, the gradients of which are respectively m1and m2. (i) If L1and L2are parallel, the angles that they make with the x axis are equal and m1= m2. e. m2 = 1. Fig. 7 *Ex 21. With the notation above, prove that, if mi.