Solid Geometry and Space Vectors
Solid geometry and space vectors
Know the axioms and positional relations of lines and planes: parallel, intersecting, skew, perpendicular, and the theorems that justify them. In space coordinates, a plane has equation ax+by+cz+d=0 with normal vector (a,b,c); a line is described by a point and direction. Use normals to compute angles between planes and distances from a point to a plane, and use projections to find the foot of a perpendicular. Volumes of prisms, pyramids, and spheres, and surface areas, should be memorized.
Traps: using the cosine of the angle between normals as the angle between planes without considering the acute or obtuse choice; using cos rather than sin for the angle between a line and a plane; forgetting the 1/3 factor in a pyramid's volume; and setting up a non-right-handed coordinate system. Diagrams are helpful, but coordinates must be justified.
Worked example: in a cube of side 1 with A(0,0,0), B(1,0,0), D(0,1,0), and A₁(0,0,1), the plane through A, B, and D has normal (1,1,1), from x+y+z=1. The angle between this plane and the base plane z=0 is the angle between normals (1,1,1) and (0,0,1), whose cosine is 1/√3. Rule of thumb: build a right-handed coordinate frame at a vertex, then translate every geometric condition into vector equations.