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Aleja

r: P

s: Q

Q'=

P'=

A

\vec{u}_r

\vec{u}_s

Cálculo distancia entre rectas

\overrightarrow{QP}

d\left( r,s  \right) =0

d\left(r,s   \right) =0

π: plano paralelo a r que contiene a s

P, punto cualquiera de r

Punto P sobre r

Q = (3, -5, 1)
Q': (5, -4, 1)
P = (-1, -3, 2)
CasillaDeEntrada1 = Textfield(P)
CasillaDeEntrada2 = Textfield(Q)
CasillaDeEntrada3 = Textfield(Q')
O = Intersect(xAxis, yAxis)
u_s = Vector(Q, Q')
s = Line(Q, u_s)
qp = Vector(Q, P)
u'1: x(u_s)
u'2: y(u_s)
u'3: z(u_s)
qp1: x(qp)
qp2: y(qp)
qp3: z(qp)
P': (1, -2, 2)
CasillaDeEntrada4 = Textfield(P')
u_r = Vector(P, P')
r = Line(P, u_r)
listau: {u'1, u'2, u'3}
listaqp: {qp1, qp2, qp3}
listav: {x(u_r), y(u_r), z(u_r)}
ran2 = MatrixRank({listau, listav})
ran3 = MatrixRank({listau, listav, listaqp})
texto5 = If(ran2 ≟ 1 ∧ ran3 ≟ 1, "Las rectas r y s son coincidentes", ran2 ≟ 1 ∧ ran3 ≟ 2, "Las rectas r y s son paralelas", ran2 ≟ 2 ∧ ran3 ≟ 2, "Las rectas r y s son secantes", ran2 ≟ 2 ∧ ran3 ≟ 3, "Las rectas r y s se cruzan")
w: u_r ⊗ u_s
E: P + w
v = Vector(P, E)
G: Q + v
p' = Plane(Q, Q', G)
p = Plane(P, E, P')
t = IntersectPath(p', p)
I = Intersect(s, t)
J = Intersect(r, t)
u1: x(u_r)
u2: y(u_r)
u3: z(u_r)
texto1: "r:" + (LaTeX[P]) + "+λ\left( " + (LaTeX[u1]) + "," + (LaTeX[u2]) + "," + (LaTeX[u3]) + "  \right) "
texto2: "s:" + (LaTeX[Q]) + "+λ\left( " + (LaTeX[u'1]) + "," + (LaTeX[u'2]) + "," + (LaTeX[u'3]) + "  \right) "
v1: x(v)
v2: y(v)
v3: z(v)
texto6: "\vec{v}=\vec{u}_r×\vec{u}_s=\left| \begin{array}{} i & j & k \\ " + (LaTeX[u1]) + " & " + (LaTeX[u2]) + " & " + (LaTeX[u3]) + "\\ " + (LaTeX[v1]) + " & " + (LaTeX[v2]) + " & " + (LaTeX[v3]) + " \\ \end{array} \right| =\left( " + (LaTeX[v1]) + "," + (LaTeX[v2]) + "," + (LaTeX[v3]) + "  \right) "
texto11 = If(x(P) < 0, ("x+" * abs(x(P))), x(P) ≟ 0, "x", x(P) > 0, ("x-" * abs(x(P))))
texto12 = If(y(P) < 0, ("y+" * abs(y(P))), y(P) ≟ 0, "y", y(P) > 0, ("y-" * abs(y(P))))
texto13 = If(z(P) < 0, ("z+" * abs(z(P))), z(P) ≟ 0, "z", z(P) > 0, ("z-" * abs(z(P))))
texto14: "π:\left| \begin{array}{} " + (LaTeX[texto11]) + " & " + (LaTeX[texto12]) + " & " + (LaTeX[texto13]) + " \\ " + (LaTeX[u1]) + " & " + (LaTeX[u2]) + " & " + (LaTeX[u3]) + " \\ " + (LaTeX[v1]) + " & " + (LaTeX[v2]) + " & " + (LaTeX[v3]) + "\\ \end{array} \right| =0⇒π:" + (LaTeX[p]) + ""
texto21 = If(x(Q) < 0, ("x+" * abs(x(Q))), x(Q) ≟ 0, "x", x(Q) > 0, ("x-" * abs(x(Q))))
texto22 = If(y(Q) < 0, ("y+" * abs(y(Q))), y(Q) ≟ 0, "y", y(Q) > 0, ("y-" * abs(y(Q))))
texto23 = If(z(Q) < 0, ("z+" * abs(z(Q))), z(Q) ≟ 0, "z", z(Q) > 0, ("z-" * abs(z(Q))))
A: Q + v
u = Vector(Q, A)
texto10: "t:\begin{cases} " + (LaTeX[p]) + "  \\  " + (LaTeX[p']) + " \end{cases}"
B = Point(r)
rp = OrthogonalLine(B, s, space)
texto15: "\vec{u}_r=\left(  " + (LaTeX[u1]) + "," + (LaTeX[u2]) + "," + (LaTeX[u3]) + " \right) "
texto16: "\vec{u}_s=\left( " + (LaTeX[u'1]) + "," + (LaTeX[u'2]) + "," + (LaTeX[u'3]) + "  \right) "
i = false
uqp: u_r ⊗ qp
nump: abs(uqp)^(2)
denp: abs(u_r)^(2)
C = Intersect(s, rp)
drsp = Segment(B, C)
disp: abs(uqp) / abs(u_r)
texto4: "d=" + disp + ""
texto7: "\left| \vec{u}_r \right| =\sqrt{" + (LaTeX[denp]) + "}"
uqp1: x(uqp)
uqp2: y(uqp)
uqp3: z(uqp)
texto8: "\vec{u}_r×\overrightarrow{QP}=\left|\begin{array}{} i & j & k \\ " + (LaTeX[u1]) + " & " + (LaTeX[u2]) + " & " + (LaTeX[u3]) + "\\ " + (LaTeX[qp1]) + " & " + (LaTeX[qp2]) + " & " + (LaTeX[qp3]) + " \\ \end{array}  \right| =\left(" + (LaTeX[uqp1]) + "," + (LaTeX[uqp2]) + "," + (LaTeX[uqp3]) + "   \right) "
texto19: "\overrightarrow{QP}=\left( " + (LaTeX[qp1]) + " ," + (LaTeX[qp2]) + "," + (LaTeX[qp3]) + " \right) "
texto9: "\left| \vec{u}_r×\overrightarrow{QP}\: \right| =\sqrt{" + (LaTeX[nump]) + "}"
texto3: "d\left(r,s  \right) =d\left( P,s  \right) =\frac{\left|\vec{u}_r×\overrightarrow{QP}\:  \right| }{\left| \vec{u}_r \right| }=\frac{\sqrt{" + (LaTeX[nump]) + "}}{\sqrt{" + (LaTeX[denp]) + "}}=" + (LaTeX[disp]) + "\:u"
s' = Line(Q, r)
b = OrthogonalPlane(P, w)
c = OrthogonalPlane(Q, w)
P1 = Point(r)
texto24: "π:\left| \begin{array}{} " + (LaTeX[texto21]) + " & " + (LaTeX[texto22]) + " & " + (LaTeX[texto23]) + " \\ " + (LaTeX[u'1]) + " & " + (LaTeX[u'2]) + " & " + (LaTeX[u'3]) + "\\ " + (LaTeX[u1]) + " & " + (LaTeX[u2]) + "& " + (LaTeX[u3]) + " \\ \end{array} \right|=0⇒π:" + (LaTeX[c]) + " "
f = OrthogonalLine(P, c)
g = OrthogonalLine(P1, c)
F = Intersect(f, c)
H = Intersect(g, c)
av: v1^(2) + v2^(2) + v3^(2)
P1_1: x(P)
P2: y(P)
P3: z(P)
D: (-((x(w) * x(Q)) + (y(w) * y(Q)) + (z(w) * z(Q))))
texto27: "d\left( r,s  \right) =d\left( P,π  \right) =\frac{\left|\left( " + (LaTeX[v1]) + "  \right) \left( " + (LaTeX[P1_1]) + "  \right) +\left( " + (LaTeX[v2]) + "  \right) \left(" + (LaTeX[P2]) + "   \right) +\left( " + (LaTeX[v3]) + "  \right) \left( " + (LaTeX[P3]) + "  \right) +\left( " + (LaTeX[D]) + "  \right)   \right| }{\sqrt{" + (LaTeX[av]) + "}}="
numc: abs((x(w) * x(P)) + (y(w) * y(P)) + (z(w) * z(P)) + D)
disc: numc / sqrt(av)
h = Segment(P, F)
texto28: "=\frac{" + (LaTeX[numc]) + "}{\sqrt{" + (LaTeX[av]) + "}}=" + (LaTeX[disc]) + "\:u"
j_1 = Segment(P1, H)
texto30: "d=" + h + ""
K: P + w
a = Plane(P', P, K)
L: Q + w
d = Plane(Q, Q', L)
k_1 = IntersectPath(a, d)
M = Intersect(k_1, r)
N = Intersect(s, k_1)
l = Segment(M, N)
j: false
γ = Angle(r, rp)
δ = Angle(s, rp)
m = OrthogonalLine(P, s, space)
R = Intersect(s, m)
n = Segment(P, R)
texto32: "" + texto4 + ""
e: false
ε = Angle(r, m)
ζ = Angle(s, m)
k: false