Let a point P move in space at speed t; = ds/dt = |v| ≠ 0. Show:b) k = v−3|v × a|.c) If w...

Let a point P move in space at speed t; = ds/dt = |v| ≠ 0. Show:

b) k = v−3|v × a|.

c) If w = da/dt and v × a ≠ 0, then τ = |v × a|−2v × a • w.

d) The path is a straight line if and only if κ = 0.

e) The path lies in a plane if and only if τ = 0.

Note. Where v = 0, the curvature κ is defined to be 0. It v = 0, the path reduces to a point.