Math 225: Calculus III	Quiz 3    2/4&2/6/96

<< "Calculus`Math225`"; i = {1, 0, 0}; j = {0, 1, 0}; k = {0, 0, 1};

1. A particle's position is given by r[t_] = Sin[π t] i + (E^(1 - t)) j + (((t + 1))^2) k.
    Find the particle's speed when it reaches the point {0,1,4}.

Solution

First solve for t such that r[t]=={0,1,4}. It is easy to see that the only solution is t == 1. The speed is the length of the particle's velocity vector:

r[t_] = Sin[π t] i + (E^(1 - t)) j + (((t + 1))^2) k;

r'[t]

v=r'[1]

Norm[v]

N[%]

2.  Find a (non-zero) vector perpendicular to the curve defined by r[t_] = t i + t j - (t^2) k  at the point {2,2,-4}.

Solution

The unit normal vector n[t_]=((T'[t])/(Norm[T'[t]]))is perpendicular to the curve. The point {2,2,-4} clearly corresponds to t == 2, so we must calculate n[2]. Actually, it is enough to  calculate T'[2] since this vector is also perpendicular to the curve.

r[t_] = t i + t j - (t^2) k

T[t_]  =  r'[t]/Norm[r'[t]] // Simplify

T'[t] // Simplify

T'[2]

Any multiple of this vector, e.g., {2c,2c,c}, for any scalar c, will be perpendicular to the curve as well.