\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard C02

Vector Functions
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At the end of the course, each student should be able to…

  • C02: VectFunc. Model curves in Euclidean space with vector functions.

C02. Vector Functions and Curves

  • A vector function maps parameters \(t\) to points/vectors \(\vect r(t)\) on a curve in 2D or 3D space.
    • Arbitrary vector functions may be sketched by the use of a chart of \(t,x,y,z\) values.
    • Vector functions may also be expressed as parametric equations defining each component \(x(t),y(t),z(t)\) in terms of \(t\).
  • Lines and line segments are given by linear vector equations.
    • The line passing through a point \(P_0\) and parallel to the vector \(\vect v\) has equation \(\vect r(t)=P_0+\vect v t\).
    • The line segment beginning at \(P_0\) and ending at \(P_1\) has equation \(\vect r(t)=P_0+(P_1-P_0)t\) with the domain \(0\leq t\leq 1\).
  • Curves defined by \(y=f(x)\) may be easily modeled by a vector function.
    • Let \(\vect r(t)=\<t,f(t)\>\) to parametrize the curve left-to-right.
    • Let \(\vect r(t)=\<-t,f(-t)\>\) to parametrize the curve right-to-left.
  • Circles in the \(xy\) plane are modeled using sine and cosine.
    • The circle with radius \(r\) and center \(P_0\) may be parametrized counter-clockwise with \(\vect r(t)=P_0+\<r\cos t,r\sin t\>\).
    • The circle with radius \(r\) and center \(P_0\) may be parametrized clockwise with \(\vect r(t)=P_0+\<r\sin t,r\cos t\>\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 11.5 (exercises 1-20)
    • 10.1 (exercises 19-26)