# 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:
1. The points of intersection between the sphere and the line
2. The area of the triangle containning the pts in (1) and the north pole of the sphere.

### 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`

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>