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 \(\vect r(t)=P_0+\<r\sin t,r\cos t\>\).
- University Calculus: Early Transcendentals (3rd Ed)
- 11.5 (exercises 1-20)
- 10.1 (exercises 19-26)