Vector Functions

• Taking variables, return a vector
• Vector functions of a single variable in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$:
  • $\vec{r}(t)=⟨f(t),g(t)⟩$
  • $\vec{r}(t)=⟨f(t),g(t),h(t)⟩$ where $f(t),g(t),h(t)$ are “component functions”
• Domain = set of all $t$‘s for which the component functions are defined

Vector Function Domain Example

$\vec{r}(t)=⟨\cos t,\ln (4-t),\sqrt{t+1}⟩$

• $\cos t$: All $t$’s
• $\ln (4-t)$: $t<4$
• $\sqrt{t+1}$: $t\ge -1$

• Graph of vectors – think of vector returned by the vector function as a position vector for points
  • Position vector $\vec{v}=⟨a,b,c⟩$ is a vector that starts at origin and goes to $(a,b,c)$
• Basically just plug values in and plot points according to the resulting position vector

Vector Function Graphing Example

$\vec{r}(t)=⟨4{\cos }_t,4\sin t,t⟩$ Since the $z$-axis is not constant, we will get a spiral about the $z$-axis Spiral revolves around axis with $t$.

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Determining the vector equation of a line segment

Let’s say we have a segment starting at $P=(x_1,y_1,z_1)$ and ending at $Q=(x_2,y_2,z_2)$. We can use these points as info. Find direction of the line with: $\vec{v}=⟨x_2-x_{1,}{y}_2-y_1,z_2-z_1⟩$ Then, using direction $\vec{v}$ and point $P$, we then have:

$$\vec{r}(t)=⟨x_1-x_2,y_2-y_1,z_2-z_1⟩+t\vec{v}$$ $$\vec{r}(t)=⟨x_1,y_{1,}{z}_1⟩+t⟨x_2-x_1,y_2-y_1,z_2-z_1⟩$$ $$\vec{r}(t)=⟨x_1,y_1,y_1⟩+t⟨x_2,y_2,z_2⟩-t⟨x_1,y_1,z_1⟩$$ $$\vec{r}(t)=(1-t)⟨x_1,y_1,z_1⟩+t⟨x_2,y_2,z_2⟩$$

Since we want the segment to start at $P$ and $Q$, we need to restrict $t$:

• Notice that $\vec{r}(0)=⟨x_1,y_1,z_1⟩$ and $\vec{r}(1)=⟨x_2,y_2,z_2⟩$
• Thus we can just restrict $0\le t\le 1$ Then, we have: $\vec{r}(t)=(1-t)⟨x_1,y_1,z_1⟩+t⟨x_2,y_2,z_2⟩,0\le t\le 1$

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2-variable Vector Functions

Graphs become surfaces.

2-variable Vector Function Example

$\vec{r}(x,y)=x\vec{i}+y\vec{j}+(x^2+y^2)\vec{k}$

Parametric form: $x=x,y=y,z=x^2+y^2$ $\therefore$ Elliptic paraboloid ($z$-axis)

Any 1-variable or 2-variable can be written in vector form:

1-variable:

• $\vec{r}(x)=x\vec{i}+f(x)\vec{j}$
• $\vec{r}(y)=h(y)\vec{i}+y\vec{j}$

2-variable:

• $\vec{r}(x,y)=x\vec{i}+y\vec{j}+g(x,y)\vec{k}$
• $\vec{r}(y,z)=g(y,z)+y\vec{j}+z\vec{k}$
• $\vec{r}(x,z)=x\vec{i}+g(x,z)\vec{j}+z\vec{k}$

Example: $y=2x^2-5z^2=x\vec{i}+(2x^2-5z^2)+z\vec{k}$