# Two Planes, angle, line...

#### 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 (x-13)/3 = (y-6)/2 = (z-0)/1 Another way to find the line of intersection is to find a direction vector "v" and a position vector "a" for this line. We have,

> 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>