## 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 ]

Link to the commands in this file

Carlos Rodriguez <carlos@math.albany.edu> Last modified: Tue Sep 24 09:39:48 EDT 1996