The Clifford algebra for the group of rigid body motions is described. Linear elements, that is points, lines and planes are identified as homogeneous elements in the algebra. In each case the action of the group of rigid motions on the linear elements is found. The relationships between these linear elements are found in terms of operations in the algebra. That is, incidence relations, the conditions for a point to lie on a line for example are found. Distance relations, like the distance between a point and a plane are found. Also the meet and join of linear elements, for example, the line determined by two planes and the plane defined by a line and a point, are found. Finally three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.