# Solutions to the Exercises...

### Exercises in Preparation for Exam1

1.- Line of intersection between the planes P1 and P2,

> P1 := y-x-z=1; P2:= x+y-z=-1;

```
P1 := y - x - z = 1

P2 := x + y - z = -1
```

> Line := solve({P1,P2},{x,y,z});
```
Line := {y = z, x = -1, z = z}
```

 This is the line y = z on the plane x = -1. 2.- Angle between the P1 and P2

> theta := convert(angle([-1,1,-1],[1,1,-1]),degrees);

```
arccos(1/3) degrees
theta := 180 -------------------
Pi
```

> evalf(",3);
```
70.6 degrees
```

 3.- Find the angle that 2i+j makes with (-1,1,1), (1,0,1), (-2,2,1).

> Ang := proc(v1,v2) evalf(convert(angle(v1,v2),degrees),3); end;

```
Ang := proc(v1,v2) evalf(convert(angle(v1,v2),degrees),3) end
```

> t1 := Ang([2,1,0],[-1,1,1]); t2 :=Ang([2,1,0],[1,0,1]); t2 :=Ang([2,1,0],[-2,2,1]);
```
t1 := 105. degrees

t2 := 50.8 degrees

t2 := 107. degrees
```

 4.- Do the vectors: i+2k-j, i+j-k, 3i+j lie on the same plane?

> volume := innerprod([1,-1,2],crossprod([1,1,-1],[3,1,0]));

```
volume := 0
```

 So yes, they do! 5.- Compute the area of the triangle PQR where, P(0,1,0), Q(2,1,0), R(1,0,1).

> p := vector([0,1,0]); q := vector([2,1,0]); r := vector([1,0,1]);

```
p := [ 0, 1, 0 ]

q := [ 2, 1, 0 ]

r := [ 1, 0, 1 ]
```

 Notice that the area of the triangle is half the area of the twisted rectangle generated by the arrows from p to q and from q to r. Thus,

> A := crossprod(q-p,r-q)/2; Area := sqrt(innerprod(A,A));

```
A := 1/2 [ 0, -2, -2 ]

1/2
Area := 2
```

 6.- Find the symmetric equations of the line through zero perpendicular to the plane z-x-y=5.

> Line := [0,0,0] + t*[-1,-1,1];

```
Line := [0, 0, 0] + t [-1, -1, 1]
```

> symLine := { x=y,y=-z};
```
symLine := {x = y, y = - z}
```

 7.- Do the line L and plane P intersect?

> L := {x=-3*t-1, y=2*t-2, z=t-1}; P := {x+y+z=3};

```
L := {x = - 3 t - 1, y = 2 t - 2, z = t - 1}

P := {x + y + z = 3}
```

> solve(L union P,{x,y,z,t});
```
```

 no output! ... i.e. NO SOLUTION!

> solve(L union P,t);

```
```

 No output again. Clearly they don't intersect.

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