Planos-Rectas
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Transcript of Planos-Rectas
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−→N P 0
V (−−→P 0P )
−→N P 0
−→N
V (−−→P 0P ) = (x − x0, y − y0, z − z0)
V (−−→P 0P ) ·
−→N = 0
(x − x0, y − y0, z − z0) · (a,b,c) = 0
a(x − x0) + b(y − y0) + c(z − z0) = 0
ax + by + cz + d = 0
−→N = (a,b,c).
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P 0(0, 0, 6), Q(0, 8, 0), R(9, 0, 0) ℜ3.
ℜ3
V (−−→RQ) =
−→A = (−9, 8, 0),
P 0, V (−−→RP 0) =
−→B = (−9, 0, 6).
A × B =
i j k
−9 8 0
−9 0 6
−→N = 6(8, 9, 12).
P (x, y, z)
−→N
P P 0.
V (−−→P P 0) ·
−→N = 0
(x − 0, y − 0, z − 6) · 6(8, 9, 12) = 0
(x, y, z − 6) · 6(8, 9, 12) = 0
8x + 9y + 12z − 72 = 0
ax + by + cz + d = 0,
−→N = (a,b,c).
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P o(x0, y0, z0)
−→V = (a,b,c).
V (−−→P 0P )
−→V . V (
−−→P 0P ) = t
−→V ,
V (−−→P 0P ) = t
−→V
(x − x0, y − y0, z − z0) = t(a,b,c)
x − x0 = ta, y − y0 = tb, z − z0 = tc
x = x0 + ta, y = y0 + tb, z = z0 + tc, t ∈ ℜ.
P (x, y, z)
x − x0
a =
y − y0
b =
z − z0
c
P 1(2, 4,−1), P 2(5, 0, 7).
−→V ,
−→V = V (
−−−→P 1P 2),
−→V = (3,−4, 8).
P 1 P = (x, y, z)
−→
V ,
V (−−→P 1P ) = t
−→V
(x − 2, y − 4, z + 1) = t(3,−4, 8)
x = 2 + 3t, y = 4 − 4t, z = −1 + 8t, t ∈ ℜ.
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z = −1 + 8t = 0
8t = 1
t = 1
8.
t = 1
8 x = 19
8 , y = 7
2, z = 0; ( 19
8 , 7
2, 0).
−→N = (4, 2,−5).
P (3,−1, 4), Q(3, 3, 2), R(1, 1,−1)
ℜ3.
5x−2y+5z−12 = 0, 2x+y−7z +11 = 0.
−→V = (4, 5,−7).
x − 3y − 2z − 4 = 0, 3x + 2y − 4z − 6 = 0.
P (−5,−3, 4) Q(−2,−3, 3).
P (3, 2,−1) Q(1,−2, 4) ℜ3.
P (3,−1, 2)
−→U = (1, 1,−2),
−→V = (−2, 3, 1).
P (2, 2,−1) Q(3, 2, 1) R(0, 1, 0)
