PLANE SEPARATION AND VECTOR ALGEBRAThis is a more detailed look at the interpretation of plane separation in terms ofcoordinates. We shall verify in detail that for each line L the set of points not on Lsatisfies the conditions in the Plane Separation Postulate.If L is a line in the coordinate plane R2, then L is defined by an equation of the form0 = g(x, y) = Ax + By + Cwhere at least on of A, B is nonzero. The two half-planes determined by L are the setswhere g(x, y) > 0 and g(x, y) < 0. We shall denote these half-planes (or sides of the lineL) by H1and H2respectively.The first thing to notice is that H1and H2are both nonempty. For each scalar k,consider the point Vk= (kA, kB). We then have g(x, y) = k(A2+B2)+C, and since at leastone of A, B is nonzero it follows that the coefficient A2+ B2is positive. Therefore we cansay that g(Vk) = g(kA, kB) will be positive if k > −C/(A2+ B2) and g(Vk) = g(kA, kB)will be negative if k < −C/(A2+ B2). Since there are infinitely values of k satisfyingeither of these inequalities, it follows that in fact both H1and H2contain infinitely manypoints.We also need to check that H1and H2are both convex; in other words, if P = (x, y)and Q = (u, v) belong to one of these half-planes and 0 < t < 1, then the point P +t(Q− P )also belongs to the same half-plane. The key to this is the following chain of identities:gP +t(Q− P )= gx+t(u− x), y+t(v− y)= Ax+t(u− x)+ By+t(v− y)=(1 − t)(Ax + By) + t(Ax + By) + C = (1 − t) · g(P ) + t · g(Q) .If P and Q lie on the same side of L, then either g(P ) and g(Q) are both positive or theyare both negative. Note that t and 1 − t are both positive in either case. If g(P ) and g(Q)are positive, then it follows thatgP + t(Q − P )= (1 − t) · g(P ) + t · g(Q)must also be positive since it is a sum of two products of positive numbers, while if g(P )and g(Q) are negative, then it follows that the expression is a sum of two products, eachwith one positive and one negative factor, and hence in this case gP + t(Q − P )mustbe negative.Finally, we need to show if P is in one half-plane and Q is in the other, then theopen segment (P Q) and the line L have a point in common. In the terms of the precedingdiscussions, this means that we can find some t such that 0 < t < 1 and gP +t(Q− P )=0.1We shall only consider the case where g(P ) < 0 < g(Q); the other case, in whichg(P ) > 0 > g(Q), can be obtained by interchanging the roles of P and Q in the argumentbelow. By the fundamental identity displayed above, we need to find a value of t such that0 = (1 − t)g(P ) + tg(Q) = g(P ) + tg(Q) − g(P).The solution to this equation ist =−g(P )g(Q) − g(P)where the denominator is positive since g(Q) > g(P ). By assumption g(P ) is negative, andtherefore the entire expression for t is positive. Furthermore, we also have 0 < −g(P ) <g(Q) − g(P ), so it also follows that t < 1. Therefore, if we take t as given above, then thepoint P + t(Q − P ) will lie on both the open segment (P Q) and the line