Finding Meteorites with Mathematics

Everyone knows that the equation ax+ by + c = 0 (1) represents a straight line. But what if you have a three-dimensional problem? Presumably a straight line in three dimensions is represented by the equation ax+ by + cz + d = 0. (2) Wrong! Equation (2) is the equation, in three dimensions, of a plane. To represent a line in three dimensions, we need two equations of the form of Equation (2); a line is the intersection of two planes and we need to give the equations of two planes to specify it. Three non-collinear points ought to define a plane, so, if we know the coordinates of three points, (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3), how can we determine the equation of the plane containing these three points? Well, we can write down the equations ax1 + by1 + cz1 + d = 0, (3) ax2 + by2 + cz2 + d = 0, (4) and ax3 + by3 + cz3 + d = 0. (5) That doesn’t seem to be enough to solve for the four un-knowns, a, b, c and d. If I want a condition for a point (x, y, z) to be in the plane, I combine Equation (2) with Equations (3), (4), and (5) to give me four equations in the three unknowns, x y z 1 x1 y1 z1 1 x2 y2 z2 1 x3 y3 z3