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Aleja

π:\:A x+By+Cz+D=0

rg (M) = rg (M') =1 ⇒ 

Sistema compatible indeterminado
con dos parámetros. 

Planos coincidentes.

Cálculo algebraico

rg (M) ≠rg (M')⇒

Sistema incompatible.

Planos paralelos

rg (M) =2 = rg (M') ⇒

Sistema compatible indeterminado
con un parámetro.

Planos secantes.

Recta intersección

rg (M) = rg (M') = 1⇒

Sistema compatible indeterminado
con dos parámetros.

Planos coincidentes

rg (M) ≠rg (M')⇒

Sistema incompatible. 

rg (M) = rg (M') = 2⇒

Sistema compatible indeterminado
con un parámetro.

Recta común a los tres planos.

Solución

Rectas

rg(M) = rg(M') =3 ⇒

Sistema compatible determinado.
Solución única.

Punto común a los tres planos.

Ver solución

Dos  planos

Dos planos

Tres planos

Tres planos

-x-y+3z- 6 = 0

π :

x +5y -2z +5 = 0

π ':

x+y +2z +1= 0

π '':

p': (0 * z) + x + (5 * y) - (2 * z) + 5 = 0
B = PointIn(p')
bz: z(B)
by: y(B)
bx: x(B)
v = OrthogonalVector(p')
c': z(v)
b': y(v)
a': x(v)
g: (a' * bx) + (b' * by) + (c' * bz)
p: (0 * z) + (-x) - y + (3 * z) - 6 = 0
A = PointIn(p)
az: z(A)
ay: y(A)
ax: x(A)
u = OrthogonalVector(p)
c: z(u)
b: y(u)
a: x(u)
f: (a * ax) + (b * ay) + (c * az)
modu: abs(u)
O: (0, 0, 0)
d: (-f)
modv: abs(v)
disOP' = Distance(O, p')
d': (-g)
listapa: {a, b, c, d}
listap'a: {a', b', c', d'}
listap: {a, b, c}
texto4: "M=\left(\begin{array}{} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " &" + (LaTeX[c]) + "\\ " + (LaTeX[a']) + " & " + (LaTeX[b']) + "&" + (LaTeX[c']) + " \\ \end{array}    \right) "
texto5: "M'=\left( \begin{array}{} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " &" + (LaTeX[c]) + "&" + (LaTeX[d]) + "\\ " + (LaTeX[a']) + " & " + (LaTeX[b']) + "&" + (LaTeX[c']) + "&" + (LaTeX[d']) + " \\ \end{array}   \right) "
listap': {a', b', c'}
rango2 = MatrixRank({listap, listap'})
rango2a = MatrixRank({listapa, listap'a})
texto6: "rg (M) = " + rango2 + ""
texto7: "rg (M') = " + rango2a + ""
num: 2
i1 = true
i2 = false
p'': (0 * z) + x + y + (2 * z) + 1 = 0
w = OrthogonalVector(p'')
a'': x(w)
b'': y(w)
c'': z(w)
C = PointIn(p'')
cx: x(C)
cy: y(C)
cz: z(C)
e: (a'' * cx) + (b'' * cy) + (c'' * cz)
d'': (-e)
texto18: "M =\left(\begin{array}{} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " & " + (LaTeX[c]) + " \\ " + (LaTeX[a']) + " & " + (LaTeX[b']) + " & " + (LaTeX[c']) + " \\ " + (LaTeX[a'']) + " & " + (LaTeX[b'']) + " & " + (LaTeX[c'']) + " \\ \end{array}   \right) "
listap'': {a'', b'', c''}
listap''a: {a'', b'', c'', d''}
rango3 = MatrixRank({listap, listap', listap''})
rango3a = MatrixRank({listapa, listap'a, listap''a})
texto20: "rg (M) = " + rango3 + ""
texto21: "rg ( M') = " + rango3a + ""
dd': d ≟ d'
dd'': d ≟ d''
d'd'': d' ≟ d''
i31 = true
i = IntersectPath(p, p'')
i23 = true
i33 = true
r = IntersectPath(p, p'')
CasillaDeEntrada1 = Textfield(textoplano)
CasillaDeEntrada2 = Textfield(textoplano2)
r2 = IntersectPath(p', p)
CasillaDeEntrada3 = Textfield(textoplano3)
texto33: "π:" + (LaTeX[textoplano]) + ""
texto2: "π':" + (LaTeX[textoplano2]) + ""
texto3: "π'':" + (LaTeX[textoplano3]) + ""
texto19: "M'=\left( \begin{array}{} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " & " + (LaTeX[c]) + " &" + (LaTeX[d]) + "\\ " + (LaTeX[a']) + " & " + (LaTeX[b']) + " & " + (LaTeX[c']) + "&" + (LaTeX[d']) + "\\ " + (LaTeX[a'']) + " & " + (LaTeX[b'']) + " & " + (LaTeX[c'']) + "&" + (LaTeX[d'']) + " \\ \end{array}  \right) "
m = IntersectPath(p'', p')
n = IntersectPath(p, p')
P = Intersect(r, m)
j = IntersectPath(p, p'')
k = IntersectPath(p'', p')
l = IntersectPath(p, p')
h = IntersectPath(p, p')
texto17: "P" + P + ""