Homework "Vec"

1. (6 points) Build a class called "vec" (we reserve the name "vector" for n-dimensional vectors) to represent 3D vectors.

   • Something like

     public class vec{
     	public double x,y,z;
     	/* your methods go here */
     }
     

   • Implement a couple of constructurs,

     //constructors:
     public vec(){ x=y=z=0; }
     public vec(double a,double b,double c){ x=a;y=b;z=c; }
     

   • Overload mathematical operators and make a "print" method (for debugging),

     //operators:
     public static vec operator*(vec v, double c){return new vec(c*v.x,c*v.y,c*v.z);}
     public static vec operator*(double c, vec v){return v*c;}
     public static vec operator+(vec u, vec v){/*...*/}
     public static vec operator-(vec u, vec v){/*...*/}
     public static vec operator-(vec u){/*...*/}
     //methods:
     public void print(string s){Write(s);WriteLine($"{x} {y} {z}");}
     public void print(){this.print("");}
     

   • Put your vec class in the file "vec.cs" and compile it into a library,

     vec.dll: vec.cs
     	mcs -target:library -out:vec.dll vec.cs
     

   • Create a "main.cs" file with the "Main" function that illustrates your implementation,

     main.exe: main.cs vec.dll
     	mcs -target:exe -out:main.exe -reference:vec.dll vec.cs
     

2. (3 points) Implement methods for dot-product, vector-product, and norm; override ToString method. Something like

   public double dot(vec other){return this.x*other.x+this.y*other.y+this.z*other.z;}
   /* and/or */
   public static double dot(vec v,vec w){return v.x*w.x+v.y*w.y+v.z*w.z;}
   

3. (1 point) Make an "approx" method to compare two vec's with absolute precision τ and relative precision ε. Something like

   static bool approx(double a,double b,double tau=1e-9,double eps=1e-9){
   	if(Abs(a-b)<tau)return true;
   	if(Abs(a-b)/(Abs(a)+Abs(b))<eps)return true;
   	return false;
   	}
   public bool approx(vec other){
   	if(!approx(this.x,other.x)return false;
   	if(!approx(this.y,other.y)return false;
   	if(!approx(this.z,other.z)return false;
   	return true;
   	}
   public static bool approx(vec u, vec v){ return u.approx(v); }
   

   and run extensive tests of your implementation in the Main function (see [../matlib/complex/main.cs] for inspiration).