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Vector-valued functions.

When a particle moves in the space, during a time-interval I, we think of its coordinates as functions of the variable t (= time), namely (x=x(t),y=y(t),z=z(t)). The set of all the locations of the particle is a curve $\mathcal{C}$ in the space, named the trajectory of the particule, and the functions x,y,z define a parametrization of $\mathcal{C}$.


  
Figure 1: The position of a particle.
\begin{figure}
\mbox{\epsfig{file=PositionVector.eps,height=4cm} }
\end{figure}

The vector $\overrightarrow{r} =x(t) \overrightarrow{i} + y(t) \overrightarrow{j} +z(t) \overrightarrow{k} $ is the position vector of the particle at time t, and the point M such that $ \overrightarrow{OM} =\overrightarrow{r} $ is the position of the particle at time t.

Without the z-component, we have the corresponding definitions for a curve/ the motion of a particule in the plane.

Example 2.1   Take

\begin{displaymath}\overrightarrow{r} = \underbrace{ \sin t }_{x(t)} \overrighta...
...e{ \cos 2t }_{y(t)} \overrightarrow{j} , \; t \in \mathbb{R} .
\end{displaymath}

We have: $\forall t \in \mathbb{R} , \; y(t)=1-2x(t)^2$, i.e. the trajectory of the particle is on the parabola whose equation is y=1-2x2. Is the trajectory the whole parabola? No! because $\forall t \in \mathbb{R} ,\; -1 \leq x(t) \leq 1$. With the notations of Figure  2 the trajectory is the arc AC, on which the particule travels alternatively in both directions, with a period of $2 \pi$.


  
Figure 2: A parabolic trajectory.
\begin{figure}
\mbox{\epsfig{file=ArcParabole.eps,height=4cm} }
\end{figure}

Example 2.2   The curve defined by $\overrightarrow{r} =\cos (3t) \overrightarrow{i} + \sin (2t) \overrightarrow{j} $ is called a ``Lissajoux curve'' (it is displayed in Figure  3).


  
Figure 3: A Lissajoux curve.
\begin{figure}
\mbox{\epsfig{file=lissajouxc3s2.eps,height=7cm} }
\end{figure}

Example 2.3   Take now

\begin{displaymath}\overrightarrow{r} = \underbrace{ \cos t}_{x(t)} \overrightar...
...\underbrace{1}_{z(t)}
\overrightarrow{k}\; t \in \mathbb{R} .
\end{displaymath}

The trajectory is the unit circle in the plane whose equation is z=1.
  
Figure 4: A circular trajectory.
\begin{figure}
\mbox{\epsfig{file=zUnitCircle.eps,height=4cm} }
\end{figure}

Example 2.4   Take now

\begin{displaymath}\overrightarrow{r} = \underbrace{ \cos t}_{x(t)} \overrightar...
...\underbrace{t}_{z(t)}
\overrightarrow{k}\; t \in \mathbb{R} .
\end{displaymath}

The trajectory is a circular helix, whose projection onto the xy-plane is the unit circle and the axis is the z-axis, as in Figure  5.
  
Figure 5: An helix.
\begin{figure}
\mbox{\epsfig{file=Helix.eps,height=8cm} }
\end{figure}


next up previous contents
Next: Limits and Continuity. Up: Curves in the plane Previous: Curves in the plane
Noah Dana-Picard
2001-05-30