By Dawkins P.
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Note that it is very easy to modify the above vector function to get a circle centered on the x or yaxis as well. For instance, = r (t ) 10sin t , −3,10 cos t will be a circle of radius 10 centered on the y-axis and at y = −3 . In other words, as long as two of the terms are a sine and a cosine (with the same coefficient) and the other is a fixed number then we will have a circle that is centered on the axis that is given by the fixed number. aspx Calculus III Let’s take a look at a modification of this.
Notice that = r ( 0) = x1 , y1 , z1 r (1) x2 , y2 , z2 So, if we restrict t to be between zero and one we will cover the line segment and we will start and end at the correct point. aspx Calculus III As noted briefly at the beginning of this section we can also vector functions of two variables. In these cases the graphs of vector function of two variables are surfaces. So, to make sure that we don’t forget that let’s work an example with that as well. Example 8 Identify the surface that is described by r ( x, y ) = x i + y j + ( x 2 + y 2 ) k .
The third component is only defined for t ≥ −1 . Putting all of these together gives the following domain. [ −1, 4 ) This is the largest possible interval for which all three components are defined. Let’s now move into looking at the graph of vector functions. In order to graph a vector function all we do is think of the vector returned by the vector function as a position vector for points on the graph. Recall that a position vector, say v = a, b, c , is a vector that starts at the origin and ends at the point ( a, b, c ) .