Here is the attachement that I need help with please show me the full solution S

Question

Here is the attachement that I need help with please show me the full solution

Show that the position vector r rightarrow (t) of a point on a circle parametrized by t is always perpendicular to the tangent to this circle at the given point by differentiating, with respect to t, the equation |r rightarrow (t)|2 = constant. Consider a function of three variables for which f (x, y, z) and (x, y, z) are always parallel: f (x, y, z) || (x, y, z). Use a half circle joining (0, 0, a) and (0.0, -a) and the previous item to show that f (0, 0, a) = f (0, 0, -a) for any number a.

Explanation / Answer

1. d / dt (|r(t)|^2) = d / dt (x(t)^2 + ... + z(t)^2) = 2x(t) * x'(t) + ... + 2z(t) * z'(t)

--> 2<x(t), y(t), z(t)> * <x'(t), y'(t), z'(t)> = d / dt (constant) = 0

--> r(t) * r'(t) = 0 --> r(t) and r'(t) are perpendicular.

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