Consider two curves y and 72 which are defined parametrically by the functions ry and ro, respectively, which in turn are defined by the rules +2 - t + 8 ri (t) = and r2(9) = +2 + t S respectively, fort, se R.(i) These two curves intersect two points in IR whose position vectors are v and w. Notice that one of these points of intersection is obtained by setting 8 = t = 0, in which case we get v = 0. The position vector of the second point of intersection is W [2;0] (ii) By considering the angles between the tangent vectors to y1 and 72 at their points of intersection, we can find the angles between y1 and 72 at these points. Let O be the angle between y1 and 72 at v. We have that cos 0 = 0 Likewise, if we let 4 be the angle between y] and 72 at w, then COS = (iii) Now, consider the standard basis vector e1 = There is one point (with position vector x1) on the curve y1 at which the tangent to the curve is parallel to e1: X1 = Likewise, there is one point (with position vector x2) on the curve 72 at which the tangent to the curve is parallel to e1: X2(iv) Similarly, consider the standard basis vector e2 (9) There is one point (with position vector y1 ) on the curve y1 at which the tangent is parallel to e2 y1 Likewise, there is one point (with position vector yo) on the curve 2 at which the tangent to the curve is parallel to e2: y2 = (v) The plots for 71 and 72 are actually rather similar. However, the graph below is a plot of the curve 71 6 5- 4 2 2 4 5 6