TITLE: Solution to Problem1:
# !b1 Problem1:
 Consider the unit sphere centered at the origin and 
   the line with direction (2,1-3a,2-3a) passing through the point
   (0,a,a). When a=sqrt(2)/2 Find:
  <ol>
    <li> The points of intersection between the sphere and the line
    <li> The area of the triangle containning the pts in (1) and
       the north pole of the sphere.
  </ol>

!b3 SOLUTION:

> with(linalg): a := sqrt(2)/2:
> x := 2*t; y:= a + (1-3*a)*t; z:= a +(2-3*a)*t;
                                   x := 2 t

                                 1/2             1/2
                       y := 1/2 2    + (1 - 3/2 2   ) t

                                 1/2             1/2
                       z := 1/2 2    + (2 - 3/2 2   ) t

> tstar := solve(x^2+y^2+z^2 = 1, t);

                                tstar := 0, 1/3

# If you wanna see the equation....
> expand(x^2+y^2+z^2 = 1);
                      2            1/2            2  1/2
                  18 t  + 1 + 3 t 2    - 6 t - 9 t  2    = 1

> sort(%,t);
                      2      1/2  2      1/2
                  18 t  - 9 2    t  + 3 2    t - 6 t + 1 = 1

> collect(%,t);
                           1/2   2            1/2
                  (18 - 9 2   ) t  + (-6 + 3 2   ) t + 1 = 1

> L := t -> vector([0,a,a]) + t*vector([2,1-3*a,2-3*a]);
                 L := t -> [0, a, a] + t [2, 1 - 3 a, 2 - 3 a]

> P1 := L(0); P2 := L(1/3); P3 := vector([0,0,1]);
                               [        1/2       1/2]
                         P1 := [0, 1/2 2   , 1/2 2   ]

            [        1/2       1/2]       [            1/2           1/2]
      P2 := [0, 1/2 2   , 1/2 2   ] + 1/3 [2, 1 - 3/2 2   , 2 - 3/2 2   ]

                                P3 := [0, 0, 1]

> evalm(P2);
                                [2/3, 1/3, 2/3]

> len := vec -> sqrt(innerprod(vec,vec));
                    len := vec -> sqrt(innerprod(vec, vec))

> area := len(crossprod(P2-P1,P3-P1))/2;
                                               1/2
                            area := 1/2 - 1/6 2

> evalf(area);
                                  .2642977396

>