1.1. LOCATING POINTS IN SPACE 17 • P v u u v α Figure 1.11: Oblique Coordinates (b) Show that the distance of a point P with oblique coordinates ( u,v ) from the origin is given by dist( P, O ) = r u 2 + v 2 + 2 | uv | cos α. (c) Show that the distance between points P (with oblique coordinates ( u 1 ,v 1 )) and Q (with oblique coordinates ( u 2 ,v 2 )) is given by dist( P,Q ) = r △ u 2 + △ v 2 + 2 △ u △ v cos α where △ u := u 2 − u 1 △ v := v 2 − v 1 . ( Hint: There are two ways to do this. One is to substitute the expressions for the rectangular coordinates in terms of the
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