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Unformatted text preview: 64 CHAPTER 1. COORDINATES AND VECTORS ℓi in Pi (i = 1, 2) through P0 and perpendicular to ℓ0 . Then θ is by
deﬁnition the angle between ℓ1 and ℓ2 .
To relate this to the equations of P1 and P2 , consider the plane P0
(through P0 ) containing the lines ℓ1 and ℓ2 . P0 is perpendicular to ℓ0 and
hence contains the arrows with tails at P0 representing the normals
→
−
→
−
→
−
→
−
→
→
→
→
N 1 = A1 − + B1 − + C1 k (resp. N 2 = A2 − + B2 − + C2 k ) to P1 (resp.
ı ı →
−
P2 ). But since N i is perpendicular to ℓi for i = 1, 2, the angle between the
→
−
→
−
vectors N 1 and N 2 is the same as the angle between ℓ1 and ℓ2
(Figure 1.34), hence
→→
−−
N1 · N2
cos θ = − −
(1.23)
→→
N1 N2
For example, the planes determined by the two equations
x
+
√ y +z = 3
x + 6y −z = 2
meet at angle θ , where
cos θ = √ (1, 1, 1) · (1, √ 6, −1)
√2
12 + 6 + (−1)2 12 + 12 + 12
√
1+ 6−1
√√
=
38
√
6
=√
26
1
=
2 so θ equals π/6 radians. Parametrization of Planes
So far, we have dealt with planes given as loci of linear equations. This is
an implicit speciﬁcation. However, there is another way to specify a plane,
which is more explicit and in closer analogy to the parametrizations we
have used to specify lines in space.
Suppose
→
→
− = v − +v − +v −
→
→
v
1ı
2
3k
→
→
− = w − +w − +w −
→
→
w
ı k
1 2 3 ...
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 Fall '08
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 Calculus, Equations, Vectors

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