Section 2.1 Vector Functions

2.1 Vector Functions

In this section, we introduce vector functions. In previous section, given a point $P=(x_1,y_1,z_1)$ and a direction $\overrightarrow{v}=(a,b,c)$, we can find the vector equation of a line, $<x,y,z>=<x_1+at,y_1+bt,z_1+ct>$. This kind of presentation can be extended to curves in the space when each component is not linear. 

Definition: 

A vector-valued function is a function of the form $\overrightarrow{r(t)}=<f(t),g(t)>$ in 2-dimensional space or $\overrightarrow{r(t)}=<f(t),g(t),h(t)>$ in $3$-dimensional space, where the component functions $f(t)$, $g(t)$, and $h(t)$, are real-valued functions of the parameter $t$. The domain of the vector function is defined by the intersection of domains of each component. 

Example 1: Find the domain of the vector function $<\text{ln}(t-1),\text{sec}(t),\sqrt{t^2-1}>$.

Exercise 1: Find the domain of the vector function $<\text{ln}(1-t^2),\text{tan}(t),\sqrt{t+1}>$.

Definition: 

(a) The limit of a vector function $\overrightarrow{r(t)}=<f(t),g(t),h(t)>$ is defined by taking limit of each component

$$\underset{t\to a}{lim}\overrightarrow{r(t)}=<\underset{t\to a}{lim}f(t),\underset{t\to a}{lim}g(t),\underset{t\to a}{lim}h(t)>.$$

(b) A vector function $\overrightarrow{r(t)}=<f(t),g(t),h(t)>$ is continue at $t=a$ if 

$$\underset{t\to a}{lim}\overrightarrow{r(t)}=\overrightarrow{r(a)}.$$

Example 2: Find the limit of $\overrightarrow{r(t)}=<e^{-3t},\frac{t}{\text{sin}t},{\text{sin}}^{-1}(2t)>$ at $t=0$. Is this vector function continue at $t=0$?

Exercise 2: Find the limit of $\overrightarrow{r(t)}=<{\text{tan}}^{-1}(t),\text{ln}(t),\frac{t-1}{{\text{cos}}^{-1}(t)}>$ at $t=1$. Is this vector function continue at $t=1$?

We learn how to draw a parameter curve, $(x,y)=(f(t),g(t))$ in 2-dimensional space, here we can do the same for $3$-dimensional space. 

Definition: The set of point all points $(x,y,z)=(f(t),g(t),h(t))$ such that $f(t)$, $g(t)$ and $h(t)$ are continuous functions with respect to the parameter $t$ is called a space curve. 

Example 3: Sketch the curve of the vector function

$$\overrightarrow{r(t)}=<3\text{sin}(t),2\text{cos}(t),1>.$$

Exercise 3: Sketch the curve of the vector function

$$\overrightarrow{r(t)}=<-2,3\text{cos}(t),4\text{sin}(t)>.$$

Example 4: Show the curve $x={\text{cos}}^2(t)$, $y=3{\text{sin}}^2(t)$ and $z=\text{cos}(t)$ is the intersection of two surfaces. Find the equations of two surfaces.

Exercise 4: Show the curve $x=\text{sin}(t)$, $y={\text{sin}}^2(t)$ and $z=2{\text{cos}}^2(t)$ is the intersection of two surfaces. Find the equations of two surfaces.

Example 5: Find a vector function that presents the intersection of two surfaces: $y^2+\frac{z^2}{2^2}=1$ and $x=1$. Sketch the curve.

Exercise 5: Find a vector function that presents the intersection of two surfaces: $\frac{x^2}{3^2}+z^2=1$ and $y=-1$. Sketch the curve.

Example 6: Find a vector function that presents the intersection of two surfaces: $y^2+z^2=9$ and $x=1+y$. Sketch the curve.

Exercise 6: Find a vector function that presents the intersection of two surfaces: $x^2+z^2=4$ and $y=z-1$. Sketch the curve.

Example 7: Sketch the curve of the vector function

$$\overrightarrow{r(t)}=<\text{sin}(t),\text{cos}(t),t>,$$

and find the equations of two surfaces such that their intersection is the curve.

Example 8: Find a vector function that presents the intersection of two surfaces: $y^2+9z^2=1$ and $x=y^2-z^2$.

Group work:

1. Sketch the curve of the vector function

$$\overrightarrow{r(t)}=<t,\text{cos}(t),\text{sin}(t)>,$$

and find the equations of two surfaces such that their intersection is the curve.

2. Find a vector function that presents the intersection of two surfaces: $4x^2+z^2=1$ and $y=x^2+z^2$.

3. Sketch the curve of the vector function

$$\overrightarrow{r(t)}=<2\text{cos}(t),2\text{cos}(t)+1,\text{sin}(t)>,$$

and find the equations of two surfaces such that their intersection is the curve.

4. Find a vector function that presents the intersection of two surfaces: $4x^2+9y^2=1$ and $z=x^2+y^2$.