(a) Show that the path x(t) = (cos t, cos t sin t, sin2 t) lies on a unit sphere. (b) V...

(a) Show that the path x(t) = (cos t, cos t sin t, sin² t) lies on a unit sphere.

(b) Verify that x(t) is always perpendicular to the velocity vector v(t).

(c) Use Proposition 1.7 to show that if a differentiable path lies on a sphere centered at the origin, then its position vector is always perpendicular to its velocity vector.