Problem:Find the angle between the planes x-y-z = 7 and x-2y+z=1.Find the symmetric equations for the line of intersection L of these planes. The two planes are: |
> P1 := x-y-z=7; P2 := x-2*y+z=1;
P1 := x - y - z = 7
P2 := x - 2 y + z = 1
| The normal vectors to these planes are: |
> n1 := vector([1,-1,-1]); n2 := vector([1,-2,1]);
n1 := [ 1, -1, -1 ]
n2 := [ 1, -2, 1 ]
| The angle "theta" between the planes is given by: |
> theta := angle(n1,n2);
1/2 1/2
theta := arccos(1/9 3 6 )
| which in degrees is approximately |
> evalf(convert(theta,degrees),3);
61.9 degrees
| The line of intersection of the two planes is: |
> L := solve({P1,P2},{x,y,z});
L := {y = 2 z + 6, x = 3 z + 13, z = z}
|
from here we can sort out things to get the symmetric equations for L |
> v := crossprod(n1,n2); a := subs(z=0,L);
v := [ -3, -2, -1 ]
a := {y = 6, x = 13, 0 = 0}
| re-writing "a" in vector form as, |
> a := vector([13,6,0]);
a := [ 13, 6, 0 ]
| we can write the vector form for the Line as, |
> Line := a + t*v;
Line := a + t v
| or in component form, |
> evalm(Line);
[ 13 - 3 t, 6 - 2 t, - t ]