The vector equations of steeply dipping veins and their intersections from drillhole data

"This paper describes the use of vector analysis for determining the intersection of mineral deposit veins. The mathematical evolution of the analysis is outlined.IntroductionMany mineral deposits can be characterized by sets of intersecting veins. Often a vein will have a steep dip and may intersect with another similar vein. Finding the intersections of these veins is usually done by graphical means. Structural geologists, by the methods of descriptive geometry, can usual ly predict where intersections of veins will occur. This method normally requires accurate drafting and layout of the problem and does not always lend itself to solution in the field.Another method for solving this problem involves the use of vector analysis, which is easily adapted to hand-held programmable calculators. It is assumed at the outset that the reader is familiar with the dot and cross products used in vector analysis. Two excellent references are cited at the end of this note for convenience.A plane, in general, is defined by three points. The points of interest here are the points where three vertical drillholes penetrate the vein, or plane, in question. The elevations of these three points are best referred to sea level. This requires knowing the elevation of the collar of the drillhole above sea level and the drillhole depth from the collar to the vein. The elevation of the intersection point is then found by subtracting the hole depth from the collar elevation. These three points now define the plane of the vein. A similar set of three points will define a second vein, and it is our object to show analytically their intersection if it exists."