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2. Velocity Vectors and Speed

If we consider the parameter t to represent time, several questions arise. If you travelled along the curve in a car whose position was X(t) , how would you feel your position changing in time? What forces would you feel when you reached some point where the curve took an interesting twist? The answers to these questions lie with the rate of change of the position vector - that is, the derivative of the vector function X(t) .

Becaue we are using vector notation rather than coordinate notation, we can use the definition for the derivative of a vector function in the plane to give the derivative of a vector function in three-space as well. We define the first derivative of the function X(t) to be the following limit, assuming that the limit exists:

    X'( t)=lim [h->0] [X(t+h)-X(t)]/h
When we think of the parameter as representing time, then the derivative X'( t) of the position vector is sometimes called the velocity vector, and is denoted V(t).

Optional

In this demo you can input a curve involving one constant c, and you will be able to observe the position vectors and the velocity vectors at different points of the curve. The default function is a twisted cubic. Draw the Car will toggle the appearance of a "car" on the curve at a given point. The car is a small blue rectangle centered at a point on the curve, with its longer axis parallel to the velocity vector. Draw the velocities toggles the rendering of velocity vectors from the minimum value of t to the position of the car. Draw the position does the same for the position vectors. Progressive curve , when used with the tapedeck, draws the curve progressively and in real time, starting from an initial point and going to a final point; it also controls the position of the car. The (min,max,# of div) type-in determines the domain and resolution for the function that you input.

Demonstration 3: Velocities and Position Vectors on Curves in Three-space
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The goal in using this demo is to familiarize oneself with the velocity vectors of a space curve and how they relate to the position vector.

    Consider the relationship of the velocity vector to the position vector for various curves - for example, a helix X(t) =(ccos(t) ,csin(t),bt) or a conical spiral X(t) =(etcos(t) ,etsin(t) ,t) .

Optional

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