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:
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.
!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 >