Unformatted text preview: 57 1.5. PLANES
6. Consider the line ℓ in the plane deﬁned as the locus of the linear
equation in the two rectangular coordinates x and y
Ax + By = C.
Deﬁne the vector −
N = A− + B − .
(a) Show that ℓ is the set of points P whose position vector −
N · p = C. →
(b) Show that if − 0 is the position vector of a speciﬁc point on the
line, then ℓ is the set of points P whose position vector −
N · ( p − p 0 ) = 0.
(c) Show that N is perpendicular to ℓ. 7. Show that if ℓ is a line given by
Ax + By = C
then the distance from a point Q(x, y ) to ℓ is given by the formula
dist(Q, ℓ) = |Ax + By − C |
A2 + B 2 (1.20) →
(Hint: Let N be the vector given in Exercise 6, and − 0 the position
vector of any point P0 on ℓ. Show that
dist(Q, ℓ) = proj− P0 Q = proj− (− − − 0 ) , and interpret this in
terms of A, B , C , x and y .) 1.5 Planes Equations of Planes
We noted earlier that the locus of a “linear” equation in the three
rectangular coordinates x, y and z
Ax + By + Cz = D (1.21) ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08