[size=150][b]Vector-valued functions[/b][/size][br][br]A function is called a [b]vector-valued function[/b] if the output of the function is a vector (instead of a real number). In particular, a vector-valued function of the following form[br][br][math]\vec{r}(t)=\langle f(t),g(t),h(t) \rangle[/math][br][br]where [math]f(t), g(t), h(t)[/math] are real-valued functions and [math]a\leq t \leq b[/math], can trace out a curve in [math]\mathbb{R}^3[/math] as [math]t[/math] varies over [math][a,b][/math] such that each point on the curve has [math]\vec{r}(t)[/math] as its position vector for some [math]t[/math]. This curve is usually called a [b]parametric curve[/b]. In other words, the curve is [b]parametrized by[/b] [math]\vec{r}(t)[/math].[br][br][br][br]

[size=150][b]Equation of a line in 3D Space[/b][/size][br][br]As we know, we can uniquely determine a line in 3D space by specifying a point on the line and the direction of the line. Suppose a line [math]L[/math] passing through the point [math]P_0=(x_0,y_0,z_0)[/math] and is in the same direction as the vector [math]\vec{v}=\langle a,b,c\rangle[/math] (this non-zero vector is called the [b]direction vector[/b] of [math]L[/math]). Let [math]P=(x,y,z)[/math] be any point on line [math]L[/math] and [math]O=(0,0,0)[/math]. We have[br][br][math]\overrightarrow{OP}=\overrightarrow{OP_0}+\overrightarrow{P_0P}[/math][br][br][math]\overrightarrow{OP_0}=\langle x_0,y_0,z_0\rangle[/math] i.e. the position vector of [math]P_0[/math].[br][br]As [math]\overrightarrow{P_0P}[/math] is parallel to [math]\vec{v}[/math], [math]\overrightarrow{P_0P}=t\vec{v}=t\langle a,b,c\rangle[/math] for some real number [math]t[/math]. Therefore, we have[br][br][math]\langle x,y,z\rangle=\langle x_0,y_0,z_0\rangle+t\langle a,b,c\rangle=\langle x_0+at,y_0+bt,z_0+ct\rangle[/math][br][br]The parametric equation of line [math]L[/math] is as follows:[br][br][math]L \ : \ \begin{cases} x & = x_0+at \\ y & = y_0+bt \\ z & = z_0+ct \end{cases}[/math][br][br]where [math]t[/math] is any real number.[br][br]Therefore, the parametric equation of a line can be regarded as the following vector-valued function:[br][br][math]\vec{r}(t)=\langle x_0+at, y_0+bt, z_0+ct\rangle[/math], [math]-\infty < t < \infty[/math][br][br]such that its parametric curve is the line passing through [math]P_0[/math] in the direction of [math]\langle a,b,c \rangle[/math].[br][br]([u]Note[/u]: If we want to parametrize a line segment instead, we need to choose a finite range of [math]t[/math]. For example, how can we parametrize a line segment from [math](1,2,5)[/math] to [math](0,-1,-3)[/math]?)[br][br][br]The line is illustrated in the applet below.

[size=150][b]Curve in 3D Space[/b][br][/size][br]In the applet below, the parametric curve is shown for a given [math]\vec{r}(t)[/math]. It is a [b]helix[/b]. You can change the functions [math]f(t), g(t), h(t)[/math] and the range of [math]t[/math] to obtain a different curve. [br][br][u]Examples[/u]:[br][br]An ellipse on the plane [math]x=2[/math]: [math]\vec{r}(t)=\langle 2, 4\cos(t), \sin(t)\rangle[/math], [math]0\leq t \leq 2\pi[/math].[br][br]A spiral on the surface of a cone: [math]\vec{r}(t)=\langle t\cos(t), t\sin(t), t\rangle[/math], [math]0\leq t \leq 6\pi[/math].[br][br]A "roller coaster" curve: [math]\vec{r}(t)=\langle \cos(t), \sin(t), 0.4\sin(2t)\rangle[/math], [math]0\leq t \leq 2\pi [/math]

[u]Remarks[/u]:[br][list=1][*]A curve can be parametrized by different vector-valued functions. For example, [math]\vec{s}(t)=\langle \cos(2t),\sin(2t),t\rangle[/math] with [math]0\leq t\leq 3\pi[/math] parametrizes the same helix in the above applet.[/*][*]When a curve is parametrized by [math]\vec{r}(t)[/math], a [b]positive orientation[/b] is given to the curve - the direction in which the curve is generated as [math]t[/math] increases from [math]a[/math] to [math]b[/math].[/*][*]If a vector-valued function gives 2D vectors as output, the function is in the following form: [math]\vec{r}(t)=\langle f(t),g(t)\rangle[/math].[/*][/list]

[b][size=150]Limit and continuity[/size][/b][br][br]The limit of a vector-valued function [math]\vec{r}(t)=\langle f(t),g(t),h(t)\rangle[/math] is defined in terms of the limit of its components as follows:[br][br][math]\lim_{t\to a}\vec{r}(t)=\langle \lim_{t\to a}f(t),\lim_{t\to a}g(t),\lim_{t\to a}h(t)\rangle[/math][br][br]That is to say, suppose [math]\lim_{t\to a}\vec{r}(t)=L[/math], where [math]L=\langle L_1,L_2,L_3\rangle[/math] is a vector in 3D space. Then we have[br][br][math]\lim_{t\to a}f(t)=L_1, \ \lim_{t\to a}g(t)=L_2, \ \lim_{t\to a}h(t)=L_3[/math].[br][br]Computing the limit of a vector-valued function boils down to computing the limits of its components. Moreover, [math]\lim_{t\to a}\vec{r}(t)[/math] exists if and only if [math]\lim_{t\to a}f(t), \ \lim_{t\to a}g(t), \ \lim_{t\to a}h(t)[/math] exist.[br][br][br]Another equivalent definition of the limit:[br][br][math]\lim_{t\to a}\vec{r}(t)=L \ \iff \ \lim_{t\to a}|\vec{r}(t)-L|=0[/math][br][br][br][u]Example[/u]: [br][br]Let [math]\vec{r}(t)=\left\langle \frac{\sin t}t, t^2-1, e^{t+2}\right\rangle[/math]. Then we have[br][br][math]\lim_{t\to 0}\vec{r}(t)=\left\langle \lim_{t\to 0}\frac{\sin t}t, \lim_{t\to 0}(t^2-1), \lim_{t\to 0}e^{t+2}\right\rangle=\langle 1,-1,e^2\rangle[/math][br][br][br][u]Definition[/u]: [math]\vec{r}(t)=\langle f(t),g(t),h(t)\rangle[/math] is said to be [b]continuous[/b] at [math] t=a[/math] if [math]\lim_{t\to a}\vec{r}(t)=\vec{r}(a)[/math] i.e. if [math]f(t),g(t),h(t)[/math] are continuous at [math]t=a[/math].[br][br][br][u]Remarks[/u]:[br][list=1][*]The definition of limit of a vector-valued function with 2D vectors as outputs can be similarly definied.[/*][*]If [math]\vec{r}(t)[/math] is continuous function of [math]t[/math], then the parametric curve it describes has no breaks or gaps, a property necessitated by the trajectory of an object.[/*][/list]