Section 2.1 Vector Functions
2.1 Vector Functions
In this section, we introduce vector functions. In previous section, given a point [latex]P=(x_{1},y_{1},z_{1})[/latex] and a direction [latex]\overrightarrow{v}=(a,b,c)[/latex], we can find the vector equation of a line, [latex]\lt x,y,z\gt =\lt x_{1}+at,y_{1}+bt,z_{1}+ct\gt[/latex]. 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 [latex]\overrightarrow{r(t)}=\lt f(t),g(t)\gt[/latex] in 2-dimensional space or [latex]\overrightarrow{r(t)}=\lt f(t),g(t),h(t)\gt[/latex] in [latex]3[/latex]-dimensional space, where the component functions [latex]f(t)[/latex], [latex]g(t)[/latex], and [latex]h(t)[/latex], are real-valued functions of the parameter [latex]t[/latex]. 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 [latex]\lt \text{ln}(t-1),\text{sec}(t),\sqrt{t^{2}-1}\gt[/latex].
Exercise 1: Find the domain of the vector function [latex]\lt \text{ln}(1-t^{2}),\text{tan}(t),\sqrt{t+1}\gt[/latex].
Definition:
(a) The limit of a vector function [latex]\overrightarrow{r(t)}=\lt f(t),g(t),h(t)\gt[/latex] is defined by taking limit of each component
\[\lim_{t\rightarrow a}\overrightarrow{r(t)}=\lt \lim_{t\rightarrow a}f(t),\lim_{t\rightarrow a}g(t),\lim_{t\rightarrow a}h(t)\gt . \]
(b) A vector function [latex]\overrightarrow{r(t)}=\lt f(t),g(t),h(t)\gt[/latex] is continue at [latex]t=a[/latex] if
\[ \lim_{t\rightarrow a}\overrightarrow{r(t)}=\overrightarrow{r(a)}. \]
Example 2: Find the limit of [latex]\overrightarrow{r(t)}=\lt e^{-3t},\frac{t}{\text{sin}t},\text{sin}^{-1}(2t)\gt[/latex] at [latex]t=0[/latex]. Is this vector function continue at [latex]t=0[/latex]?
Exercise 2: Find the limit of [latex]\overrightarrow{r(t)}=\lt \text{tan}^{-1}(t),\text{ln}(t),\frac{t-1}{\text{cos}^{-1}(t)}\gt[/latex] at [latex]t=1[/latex]. Is this vector function continue at [latex]t=1[/latex]?
We learn how to draw a parameter curve, [latex](x,y)=(f(t),g(t))[/latex] in 2-dimensional space, here we can do the same for [latex]3[/latex]-dimensional space.
Definition: The set of point all points [latex](x,y,z)=(f(t),g(t),h(t))[/latex] such that [latex]f(t)[/latex], [latex]g(t)[/latex] and [latex]h(t)[/latex] are continuous functions with respect to the parameter [latex]t[/latex] is called a space curve.
Example 3: Sketch the curve of the vector function
\[\overrightarrow{r(t)}=\lt 3\text{sin}(t),2\text{cos}(t),1\gt .\]
Exercise 3: Sketch the curve of the vector function
\[\overrightarrow{r(t)}=\lt -2,3\text{cos}(t),4\text{sin}(t)\gt . \]
Example 4: Show the curve [latex]x=\text{cos}^{2}(t)[/latex], [latex]y=3\text{sin}^{2}(t)[/latex] and [latex]z=\text{cos}(t)[/latex] is the intersection of two surfaces. Find the equations of two surfaces.
Exercise 4: Show the curve [latex]x=\text{sin}(t)[/latex], [latex]y=\text{sin}^{2}(t)[/latex] and [latex]z=2\text{cos}^{2}(t)[/latex] 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: [latex]y^{2}+\frac{z^{2}}{2^{2}}=1[/latex] and [latex]x=1[/latex]. Sketch the curve.
Exercise 5: Find a vector function that presents the intersection of two surfaces: [latex]\frac{x^{2}}{3^{2}}+z^{2}=1[/latex] and [latex]y=-1[/latex]. Sketch the curve.
Example 6: Find a vector function that presents the intersection of two surfaces: [latex]y^{2}+z^{2}=9[/latex] and [latex]x=1+y[/latex]. Sketch the curve.
Exercise 6: Find a vector function that presents the intersection of two surfaces: [latex]x^{2}+z^{2}=4[/latex] and [latex]y=z-1[/latex]. Sketch the curve.
Example 7: Sketch the curve of the vector function
\[\overrightarrow{r(t)}=\lt \text{sin}(t),\text{cos}(t),t\gt , \]
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: [latex]y^{2}+9z^{2}=1[/latex] and [latex]x=y^{2}-z^{2}[/latex].
Group work:
1. Sketch the curve of the vector function
\[\overrightarrow{r(t)}=\lt t,\text{cos}(t),\text{sin}(t)\gt ,\]
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: [latex]4x^{2}+z^{2}=1[/latex] and [latex]y=x^{2}+z^{2}[/latex].
3. Sketch the curve of the vector function
\[\overrightarrow{r(t)}=\lt 2\text{cos}(t),2\text{cos}(t)+1,\text{sin}(t)\gt , \]
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: [latex]4x^{2}+9y^{2}=1[/latex] and [latex]z=x^{2}+y^{2}[/latex].