MATH FOR ENGINEERING ANALYSIS I

MATHEMATICA  ASSIGNMENT I			DUE DATE 9/30/02

This is a group project. You can work  in a group of  3 or 4.  Please submit  one copy of the solution. Make sure you have written  the names of the group  members. Each group member must sign her/his name.

1. 	Use the command ParameticPlot[{x,y},{t,a,b}]  to plot the graphs of the curves 
  represented by the following   parametirc equations. Use a reasonable interval for t to get a complete graph.
  Note: Before you plot graphs of functions, it is recommended that you first   evaluate the command 
  Needs["Graphics`Master`"]
  
	a)	Cycloid      ({{x = t + sin t}, {y = 1 - cos t}})			b)  Astroid       ({{x = 2 (cos^2) t }, {y = 2 (sin^3) t}})

2.	The curve H(a, b) defined by the parametric equations 
		({{x = (a - b) cos t + b cos(((a - b)/b)) t}, {y = (a - b) sin t - bsin(((a - b)/b)) t}})
	is called a Hypocycloid. Using largeintervals for t, plot the hypocycloids H(8, 3),  H(8, 7),  and   H(24, 7).

3.	The curve E(a, b) defined by the parametric equations 
		({{x = (a + b) cos t - b cos(((a + b)/b)) t}, {y = (a + b) sin t - bsin(((a + b)/b)) t}})
	is called a Epicycloid. Using large intervals for t, plot the epicycloids E(8, 3),  E(24, 3),  and   E(24, 7).

4.	Use the command ParameticPlot3D[{x,y,z},{t,a,b}]  to plot the graphs of the curves 
 represented by the following   parametirc equations.
 
	a)	({{x=t               }, {y = ((-(Sqrt[3]))/2) (t^2)}, {z = (1/2) (t^(2       ))}})		b)	({{x = -(Sqrt[2]) sin t}, {y = 2  cos t       }, {z = (Sqrt[2]) sin t   }})

5.	The dot product of vectors a and b is obtained by  a.b  and the length of a is obtained by using the command Sqrt[a.a]. In order to calculate the cross product of two vectors we need a package, called  Calculus`VectorAnalysis`.  It is recommended that you  load this package before starting your work.    The command to load  this package is  Needs["Calculus`VectorAnalysis`"].  To compute the cross product u × v of  u  and  v we type Cross[u,v]. 
 Let u=-i+3j+4k, v=5i+4j-6k, and  w=i -j+2k  vectors in space.
 Find each of the following.
 	a)	6u-4v		b)	||u + v||  	c)	u . v			d)	u×v
 	e)	The angle between u and v 	f)	The volume of the parallelpiped whose edges are u, v, w.