> # hw 5\
# 11.5\
with(linalg):
Warning: new definition for   norm
Warning: new definition for   trace

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> # 45 These planes are parallel since the angle is 180 degrees -- you can also check with crossprod\
n1:=[2,4,-2];n2:=[-3,-6,3];\
costheta:=dotprod(n1,n2)/(norm(n1,2)*norm(n2,2));\
theta:=arccos(costheta);

                                n1 := [2, 4, -2]

                                n2 := [-3, -6, 3]

                                 costheta := -1

                                   theta := Pi
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> # 49 add z=z to the list below\
solve({z=x+y,2*x-5*y-z=1},{x,y});

                       {y = 1/7 z - 1/7, x = 6/7 z + 1/7}
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> # or\
solve({z=x+y,2*x-5*y-z=1},{x,y,z});

                        {z = 7 y + 1, x = 6 y + 1, y = y}
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> # 53\
directionofline:=crossprod([1,1,-1],[2,-1,3]);\
pointonline:=solve({x+y-z=2,2*x-y+3*z=1,z=0});

                        directionofline := [ 2, -5, -3 ]

                      pointonline := {z = 0, y = 1, x = 1}
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> n:=crossprod(directionofline,add([1,1,0],[-1,2,1],1,-1));\
eqn:=dotprod(n,add([x,y,z],[-1,2,1],1,-1))=0;

                                n := [ 2, -4, 8 ]

                         eqn := 2 x + 2 - 4 y + 8 z = 0
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> #57\
direction:=crossprod([1,1,1],[1,-1,2]);\
eqn:=[x,y,z]=add([0,1,2],scalarmul(direction,t));

                           direction := [ 3, -1, -2 ]

                   eqn := [x, y, z] = [ 3 t, 1 - t, 2 - 2 t ]
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> #61\
p:=[1,2,3];q:=[2,2,0];r:=[3,-1,5];\
a:=add(r,q,1,-1); b:=add(p,q,1,-1);\
norm(crossprod(a,b),2)/norm(a,2);

                                 p := [1, 2, 3]

                                 q := [2, 2, 0]

                                 r := [3, -1, 5]

                                a := [ 1, -3, 5 ]

                                b := [ -1, 0, 3 ]

                                        1/2   1/2
                                1/35 154    35
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> #65 find point on plane1 and then use distance formula\
solve({z=x+2*y+1,x=0,y=0},{x,y,z});\


                              {x = 0, y = 0, z = 1}
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> abs( 3*0+6*0-3*1-4)/sqrt(3^2+6^2+3^2);

                                          1/2
                                    7/18 6