# MA 227 Standard C02

Vector Functions

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)