7. Let u and v be two orthogonal unit vectors in R3. So we know luv 1 and u-u =

Question

7. Let u and v be two orthogonal unit vectors in R3. So we know luv 1 and u-u = 0, Define (t) = cos(t)u + sin(t)u (a) Calculate the velocity sector (b) Show that 'L (c) Calculate the acceleration vector ". (d) What is the relation between the acceleration and position vectors? 8. Here is a parametrization of a helix: o(t)-(R cos(wt), R sin(wt), kt) For definiteness, suppose that all parameters R, w, and k are positive. Think of this parametrized curve as the trajectory of a particle. How fast is the particle moving at a given time t?

Explanation / Answer

alpha(t) =cos (t)u +sin(t)v

a. alpha'(t) = -sin(t)u+cos (t)v

b. If alpha'(t) and alpha(t) are perpendicular, then their dot product must be zero.

-sin(t) cos(t) + sin(t)cos(t) =0

c. alpha"(t) =-cos(t)u-sin(t)v

d. alpha"(t) =-(cos(t)u +sin(t)v) = -alpha(t)

Therefore alpha"(t) =-alpha(t)

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