5. Vectors

e. Parametric Curves

A curve (also called a planar curve) may be specified in one of three ways:

as the graph of a function of \(1\) variable: \(y=f(x)\).
as the graph of an equation in \(2\) variables: \(F(x,y)=0\).
as a parametric curve: \(\vec r(t)=\left\langle x(t),y(t)\right\rangle\).

So far we have discussed parametric lines. Now we introduce parametric curves in general. They will be discussed in much more detail in the full chapter on Parametric Curves.

3. Parametric Curves

A parametric curve is a vector function \[ \vec r(t)=\left\langle x(t),y(t)\right\rangle \] which gives the position of a point as a function of a parameter \(t\). It can also be written as \(2\) component equations: \[ x=x(t) \qquad y=y(t) \] A graph of the parametric curve is a plot of the points \((x(t),y(t))\) for some range of \(t\).

If we take the parameter \(t\) to represent time, then \(\vec r(t)\) gives the position of a particle at time \(t\).

For example, the parabolic trajectory of a baseball could be given by: \[ \vec r(t)=\langle 36t,48t-16t^2\rangle \] or: \[ x=36t \qquad y=48t-16t^2 \] Its graph is shown.

Although the parameter is frequently the time, it can represent many other quantities, in which case the letter is frequently changed to match. A common parameter is the angle \(\theta\) from polar coordinates.

A circle of radius \(2\) can be parametrized as: \[ \vec r(\theta)=\left\langle 2\cos\theta,2\sin\theta\right\rangle \] and graphed as shown. The components are just polar coordinates \((x,y)=(r\cos\theta,r\sin\theta)\) with \(r=2\).

A more complicated “polar” curve is the starfish: \[ \vec r(\theta) =\left\langle (3+\sin(5\theta))\cos\theta,(3+\sin(5\theta))\sin\theta\right\rangle \] which may be graphed as shown.

The reason it is called a polar curve is that if we compare the parametrization with the polar coordinates \((x,y)=(r\cos\theta,r\sin\theta)\) then we see the curve can be specified in terms of polar coordinates as \(r=3+\sin(5\theta)\); as \(\theta\) goes around the circle from \(0\) to \(2\pi\), the radius \(r\) oscillates \(5\) times between \(2\) and \(4\).

Plot the line \(\vec r(t)=\langle 1+2t,3-t\rangle\).

Plot the starting point \(P=(1,3)\) and the direction vector \(\vec v=\langle2,-1\rangle\) located at \(P\). Then draw the line.

We plot the starting point \(P=(1,3)\) and then the direction vector \(\vec v=\langle2,-1\rangle\) located at \(P\). Finally, we draw a line through \(P\) in the direction \(\vec v\)..

5. Vectors

e. Parametric Curves

3. Parametric Curves

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