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#<h3> Exercises in Preparation for Exam1</h3>
1.- Line of intersection between the planes P1 and P2,
> P1 := y-x-z=1; P2:= x+y-z=-1; 

                              P1 := y - x - z = 1

                              P2 := x + y - z = -1

> Line := solve({P1,P2},{x,y,z});

                         Line := {y = z, x = -1, z = z}
# This is the line y = z on the plane x = -1.
<hr>
2.- Angle between the P1 and P2

> theta := convert(angle([-1,1,-1],[1,1,-1]),degrees);

                                     arccos(1/3) degrees
                        theta := 180 -------------------
                                              Pi

> evalf(",3);

                                  70.6 degrees
#<hr>
3.- Find the angle that 2i+j makes with<br>
 (-1,1,1), (1,0,1), (-2,2,1).
> Ang := proc(v1,v2) evalf(convert(angle(v1,v2),degrees),3); end;

Ang := proc(v1,v2) evalf(convert(angle(v1,v2),degrees),3) end

> t1 := Ang([2,1,0],[-1,1,1]); t2 :=Ang([2,1,0],[1,0,1]); t2 :=Ang([2,1,0],[-2,2,1]);

                               t1 := 105. degrees

                               t2 := 50.8 degrees

                               t2 := 107. degrees
#<hr>
4.- Do the vectors:<br>
i+2k-j, i+j-k, 3i+j<br>
lie on the same plane?

> volume := innerprod([1,-1,2],crossprod([1,1,-1],[3,1,0]));

                                  volume := 0
# So yes, they do!
<hr>
5.- Compute the area of the triangle PQR where,<br>
P(0,1,0), Q(2,1,0), R(1,0,1).
> p := vector([0,1,0]); q := vector([2,1,0]); r := vector([1,0,1]);

                                p := [ 0, 1, 0 ]

                                q := [ 2, 1, 0 ]

                                r := [ 1, 0, 1 ]

# Notice that the area of the triangle is half the area of the twisted 
rectangle generated by the arrows from p to q and from q to r. Thus,
> A := crossprod(q-p,r-q)/2; Area := sqrt(innerprod(A,A));

                             A := 1/2 [ 0, -2, -2 ]

                                           1/2
                                  Area := 2
#<hr>
6.- Find the symmetric equations of the line through zero perpendicular to the
plane z-x-y=5.

> Line := [0,0,0] + t*[-1,-1,1];

                       Line := [0, 0, 0] + t [-1, -1, 1]

> symLine := { x=y,y=-z};

                          symLine := {x = y, y = - z}
#<hr>
7.- Do the line L and plane P intersect?

> L := {x=-3*t-1, y=2*t-2, z=t-1}; P := {x+y+z=3};

                  L := {x = - 3 t - 1, y = 2 t - 2, z = t - 1}

                              P := {x + y + z = 3}

> solve(L union P,{x,y,z,t});
#no output! ... i.e. NO SOLUTION!
> solve(L union P,t);
# No output again. Clearly they don't intersect.
>