Let A(-2, -1, 3), B(0, 1, 4) be two point and

(d): x = -t-2, y = t+1, z = -t-1 be a line.

Find coordinates point M lies on (d) so that area of the triangle ABM equal to 3*sqrt(5).

This is my code

restart; with(LinearAlgebra):

A:=: B:=: M:=:

u:=A-B: v:=A-M:

T:=CrossProduct(u,v):

sol:=solve(1/2*Norm(T,2)=3*sqrt(5),{t});

for i from 1 to 2 do print('M'[i]=(subs(op(sol[i]), M[1]), subs(op(sol[i]), M[2]),subs(op(sol[i]), M[3]))) end do;

Let A(1,0,0) be a point and two lines

d1: (x-5)/3 = (y-1)/1 = (z - 2)/2,

d2: (x-5)/1 = (y-1)/1 = (z - 3)/3

Let B and C be two point lies on d1 and d2, respectively so that the three points A, B, C

Problem. Write the equation of the plane passing the point H(1,1,1) and cut the coordinates axes Ox, Oy, Oz at A, B, C respectively so that H is centre of the circumscribed of the trianlge ABC.

This is my code.

restart:

with(geom3d):

with(LinearAlgebra):

A:=:

B:=:

C:=:

H:=:

f:=(x,y,z)->x/a + y/b +z/c-1:

solve([f(H[1],H[2],H[3]) = 0, Norm(H - A, 2) = Norm(H - B, 2), Norm(H - A, 2) = Norm(H - C, 2)],{a,b,c}): assign(%):

Problem. Write the equation of the plane passing the point H(2, 1, 1) and cuts the coordinates Ox, oy, oz at A, B, C respectively so that H is orthocenter of the triangle.

This is my code.

restart:

with(geom3d):

with(LinearAlgebra):

A:= <a,0,0>:

B:=<0,b,0>:

C:=<0,0,c>:

H:=<2,1,1>:

f:=(x,y,z)->x/a + y/b +z/c-1:

eq:=solve([f(H[1],H[2],H[3]) = 0,DotProduct(B-C, A-H, conjugate = false...

My problem: Write the equation of the line (Delta) passing the point A(-2;3;4), perpendicular to the line

(l): x = -t + 1, y = 2t, z = 3t + 1

and have a distance from the point B(1;2;-4) obtain

a) maximum;

b) minimum.

Can you write for me a code? Thank you.