Exercises in Preparation for Exam11.- 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 := {y = z, x = -1, z = z}
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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
70.6 degrees
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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 := 105. degrees
t2 := 50.8 degrees
t2 := 107. degrees
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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
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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
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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}
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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}
| no output! ... i.e. NO SOLUTION! |
> solve(L union P,t);
| No output again. Clearly they don't intersect. |