Turning Points |
Caveat.Under the strictest application of geometric axioms,distances and vectors – and their equations – cease to work. Here, we temporarily set aside strict geometric axioms, We have a turning line bearing two points, A and B, at distances x_{A} and x_{B} from a fixed, arbitrary point, Q, also on the line. The motions of A and B are described by co-planar vectors, v_{A} and v_{B}, respectively. These motions are resolved into components, v_{Ar} and v_{Br}, at right angles to the turning line, and components, v_{Ap} and v_{Bp}, parallel to (actually in) the line. Point, T, around which the line is turning as a result of the co-planar motions of the points A and B, is found at the place on the turning line at which the join of the vector-tips of the components at right angles to the line, v_{Ar} and v_{Br}, cuts that line. The velocity of T has the same orientation as the turning line, and its vector, v_{Tp}, lies between T, and the place on the turning line at which the join of the tips of the original, unresolved vectors, v_{A} and v_{B}, cuts that line. We postpone discussion of how vectors might come to be disposed about a line in the co-planar way indicated here, noting now only that, if the vectors are not co-planar, but skew instead, then neither one of the joins of the vector tips cuts the turning line in a point—there is no T, and no v_{Tp}. That is, if the vectors are skew, there is no turning point. |