newt7sol

Targil 7 - mupad questions - solution notes

Question 1a

q1a:=proc(p1,p2,p3,p4)
   local V;
   begin
      V:=abs(linalg::scalarProduct( p2-p1, linalg::crossProduct( p3-p1, p4-p1 ))/6);
      return(V);
   end_proc

p1:=matrix(3,1,[0,0,0]); p2:=matrix(3,1,[1,0,0]); p3:=matrix(3,1,[0,1,0]); p4:=matrix(3,1,[0,0,1]);

matrix([[0], [0], [0]])
matrix([[1], [0], [0]])
matrix([[0], [1], [0]])
matrix([[0], [0], [1]])

Example

q1a(p1,p2,p3,p4)

Plot

plot( plot::Line3d(p1,p2) , plot::Line3d(p1,p3) , plot::Line3d(p1,p4) ,
plot::Line3d(p2,p3) , plot::Line3d(p2,p4) , plot::Line3d(p3,p4) )

MuPAD graphics

Question 1b

Take the planes to be written as equations for x,y,z (see example below)
Note the "solve" statement below gives something of the form [x=.. , y=.., z=..]
- change this to the form of a vector

q1b:=proc(P1,P2,P3)
   local p;
   begin
      p:=op(solve({P1,P2,P3},{x,y,z}));
      p:=matrix(3,1,[op(op(p,1),2), op(op(p,2),2), op(op(p,3),2)]);
      return(p);
   end_proc

Example

q1b( 2*x+y=3, x+3*z=1, x+y-z=4 )

matrix([[-2], [7], [1]])

Plot (I rotated the output)

plot( plot::Plane(matrix(3,1,[0,3,0]), matrix(3,1,[2,1,0]), Color=RGB::Red ),
      plot::Plane(matrix(3,1,[1,0,0]), matrix(3,1,[1,0,3]), Color=RGB::Green ),
      plot::Plane(matrix(3,1,[4,0,0]), matrix(3,1,[1,1,-1]), Color=RGB::Blue ),
      plot::Point3d(matrix(3,1,[-2,7,1]), Color=RGB::Black) )

MuPAD graphics

Question 1c

combine the last two

q1c:=proc(P1,P2,P3,P4)
   local V;
   begin
     V:=q1a( q1b(P1,P2,P3), q1b(P1,P2,P4), q1b(P1,P3,P4), q1b(P2,P3,P4) );
   end_proc

Example

q1c( 2*x+y=3, x+3*z=1, x+y-z=4, x-y-3*z=-2)

25/27

Questions 2a and 2b - same as 2c but remove all the z coordinates!

Question 2c
take the 2 points as vectors, and the plane as an equation of the form
ax+by+cz+d=0

q2a:=proc(p1,p2,P)
    local s1,s2;
    begin
    s1:=subs( P, x=p1[1], y=p1[2], z=p1[3]);
    s2:=subs( P, x=p2[1], y=p2[2], z=p2[3]);
    if s1*s2>0 then
       return(+1);
    elif s1*s2<0 then
       return(-1);
    else
       return(0);
    end_if;
end_proc

Example: same side

q2a( matrix(3,1,[1,0,0]) , matrix(3,1,[2,0,0]), x )

Example: opposite sides

q2a( matrix(3,1,[1,0,0]) , matrix(3,1,[-1,0,0]), x )

Question 2d
l1=list of points, l2=list of planes
answers: 1 if "yes", 0 if "no"

q2d:=proc(l1,l2)
   local i,j,k,s,t;
   begin
   s:=1;
   for i from 1 to nops(l1) do
       for j from i+1 to nops(l1) do
           t:=0;
           for k from 1 to nops(l2) do
                if q2a(l1[i],l1[j],l2[k])=-1 then
                   t:=1;
                end_if;
           end_for;
           if t<>1 then
                s:=0;
           end_if;
       end_for;
    end_for;
    return(s);
end_proc

Examples

l1:=[ matrix(3,1,[1,1,1]), matrix(3,1,[1,1,-1]), matrix(3,1,[1,-1,1]), matrix(3,1,[-1,1,1]),
matrix(3,1,[1,-1,-1]), matrix(3,1,[-1,1,-1]), matrix(3,1,[-1,-1,1]), matrix(3,1,[-1,-1,-1]) ]

[matrix([[1], [1], [1]]), matrix([[1], [1], [-1]]), matrix([[1], [-1], [1]]), matrix([[-1], [1], [1]]), matrix([[1], [-1], [-1]]), matrix([[-1], [1], [-1]]), matrix([[-1], [-1], [1]]), matrix([[-1], [-1], [-1]])]

l2:=[x,y,z]

[x, y, z]

this l2 is the 3 planes x=0, y=0, z=0

q2d(l1,l2)

l1:=[ matrix(3,1,[1,1,1]), matrix(3,1,[1,1,2])]

[matrix([[1], [1], [1]]), matrix([[1], [1], [2]])]

q2d(l1,l2)

Question 3

q3:=proc(a1,b1,r1,a2,b2,r2)
local d,i;
begin
d:=sqrt( (a1-a2)^2 + (b1-b2)^2 );
if d+r2<r1 then
   i:=1;
elif d+r1<r2 then
   i:=2;
elif d+r1=r2 or d+r2=r1 or r1+r2=d then
   i:=3;
elif d<r1+r2 then
   i:=4;
else
   i:=5;
end_if;
return(i)
end_proc