Planes, lines and points

Assume you have a plane defined by P1, P2, P3
P1( x1=2.1  yy1=5.5     zz1= 4.3  )
P2( x2=1.2  yy2=1.53   zz2= 2.54 )
P1( x3=3.9  yy3=3.5     zz3= 9.2  )

Plane xyz = a*x+b*y+c*z+d=0

    | yy1 zz1 1 |          | x1 zz1 1 |        | x1 yy1 1 |
a=  | yy2 zz2 1 |      b= -| x2 zz2 1 |    c=  | x2 yy2 1 |
    | yy3  zz3 1 |         | x3 zz3 1 |        | x3 yy3 1 |

d= -(a*x1+b*yy1+c*zz1) 
 
a = -22.973    b = 1.242   c = 9.946   d = 2.29445

[ (x1-x3), (yy1-yy3), (zz1-zz3)] =| s1 |

[ (x2-x3), (yy2-yy3), (zz2-zz3)] =| s2 |

[ (x4-x3), (yy4-yy3), (zz4-zz3)] =| s3 |

| s1 | dot | s2 | = 33.55

| s1 | dot | s2 |/(abs(s1)*abs(s2)) = t7 = 0.8055

cos(t8)=t7

t8=36.34 degrees

assume x4=5

s1 dot s3 = 0    s2 dot s3 = 0  solve for y3 and y4

yy4=3.441   zz4= 8.772

Find distance P4 to plane

abs(a*x4+b*yy4+c*zz4+d)/sqrt(a^2+b^2+c^2) = g = 1.182

sqrt( (x4-x3)^2+(yy4-yy3)^2+(zz4-zz3)^2) = g

Given a,b,c,d   and x4, yy4, zz4, find P3

l = (x3-x4)/g   = l = -0.9307

m = (yy3-yy4)/g = m = -0.0503

n = (zz3-zz4)/g = n = -0.3624

x3=g * l + x4
y3=g * m + yy4
z3=g * n + zz4

=========
TI-89 program

sqrt is square root symbol
|> is right triangle
:cos-1( ) is inverse cos

:zz()
:Prgm
:ClrIO
:setMode("Angle","Degree")
:setMode("DisplayDigits","Float 4")
:setMode("Exact/Approx","Approximate")
;DelVar yy4,zz4
:2.1->x1
:5.5->yy1
:4.3->zz1
:1.2->x2
:1.53->yy2
:2.54->zz2
:3.9->x3
:3.5->yy3
:9.2->zz3
:5->x4
:[[yy1,zz1,1][yy2,zz2,1][yy3,zz3,1]]->t1
:[[x1,  zz1,1][x2,  zz2,1][x3, zz3,1]]->t2
:[[x1,  yy1,1][x2, yy2,1][x3, yy3,1]]->t3
:det(t1)->a
:-det(t2)->b
:det(t3)->c
:a*x1+b*yy1+c*zz1->t
:-t->d
:Disp "Plane"
:Disp "a,b",{a,b}
:Disp "c,d",{c,d}
:Pause
:ClrIO
:[[x1-x3,yy1-yy3,zz1-zz3]]->s1
:[[x2-x3,yy2-yy3,zz2-zz3]]->s2
:[[x4-x3,yy4-yy3,zz4-zz3]]->s3
:det(subMat(s1,1,1,1,1))->t1
:det(subMat(s1,1,2,1,2))->t2
:det(subMat(s1,1,3,1,3))->t3
:sqrt(t1^2+t2^2+t3^2)->t4
:det(subMat(s2,1,1,1,1))->t1
:det(subMat(s2,1,2,1,2))->t2
:det(subMat(s2,1,3,1,3))->t3
:sqrt(t1^2+t2^2+t3^2)->t5
:dotP(s1,s2)->t6
:t6/(t4*t5)->t7
:cos-1(t7)->t8
:Disp "t8  angle",t8
:Pause
:ClrIO
:cross(s1,s2)->t9
:Disp "t9",t9
:Pause
:ClrIO
:sqrt(a^2+b^2+c^2)->t10
:DelVar x,y,z
:[[x ,y, z][x1 ,yy1,zz1,1][x2 ,yy2,zz2,1][x3 ,yy3,zz3,1]->tt
:Disp "tt",tt
:Pause
:ClrIO
:det(tt)->ttt
:expand(ttt)->te
:Disp "te",te
:Pause
:ClrIO
:dotP(s1,s3)->xz9
:dotP(s2,s3)->xz10
:solve(xz9=0 and xz10=0,{yy4,zz4})->t33
:exp|>list(t33),{yy4,zz4})->tz
:subMat(tz,1,1,1,1)->ty
:det(ty)->yy4
:subMat(tz,1,2,1,2)->ty
:det(ty)->zz4
:Disp "x4,yy4,zz4",{x4,yy4,zz4}
:Pause
:ClrIO
:sqrt( (x4-x3)^2 + (yy4-yy3)^2 + (zz4-zz3)^2)->g
:Disp "g point to plane",g
:(x4  -x3  )/g->l
:(yy4-yy3)/g->m
:(zz4-zz3)/g->n
:Disp "l,m,n direction",{l,m,n}
:EndPrgm