Lines
Vector Parametrizations
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Example: Parametrize the line that passes through the point P (1,-1,2)
and has direction vector 2i-3j+k
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Lines
Scalar Parametric Equations
Symmetric Form
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Example: Write a parametric equations for the line that passes through
the point P (1,-1,2) and has direction vector 2i-3j+k
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Intersecting Lines, Parallel Lines
Two distinct lines
l1 : r(t) = r0 + td,
l2 : R(u) = R0 + uD
intersect iff there are numbers t and u at which
r(t) = R(u).
Example: Find the point at which the lines




 

l1 : r (t )  (i  6 j  2k )  t (i  2 j  k ),


 
 
l2 : R(u )  (4 j  k )  u (2i  j  2k )
Intersect.
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Example: (1) Find the angle between the lines






 
 
 

l1 : r (t )  (i  6 j  2k )  t (i  2 j  k ), l2 : R(u )  (4 j  k )  u (2i  j  2k )
(2) Find the parametrization for the line that passes through their
intersection and is perpendicular to both l1 and l2.
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Lines
Distance from a Point to a Line
Let P0 be a point on l and let d be a direction vector for l. With P0 and Q as
shown in the figure, you can see that
d  P1 , l   d  P1 , Q   P0 P1 sin 
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Planes
Scalar Equation of a Plane
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Planes
Vector Equation of a Plane
We can write the equation of a plane entirely in vector notation. Set
N  Ai  Bj  Ck,
r0  OP   x0 , y0 , z0  ,
r  OQ   x, y, z 
Since
r0 = x0i + y0j + z0k
and
r = xi + yj + zk,
(13.6.1) can be written
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Planes
Unit Normals
If N is normal to a given plane, then all other normals to that plane are parallel to N
and hence scalar multiples of N. In particular there are two normals of length 1:
uN 
N
N
and
 uN 
N
N
Intersecting Planes
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
p1 : 2( x  1)  3 y  5( z  2)  0, p2 : 4 x  6 y  10z  24,
p3 : 4 x  6 y  10z  1  0,
p4 : 2 x  3 y  5 z  12  0
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Example: Show that the plane
p1 : 2 x  3 y  2 z  9,
p2 : x  2 y  z  4
are non-parallel and find a scalar parametric equations for the line
formed by the two intersecting planes.
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Planes
The Plane Determined by Three Noncollinear Points
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Planes
The Distance from a Point to a Plane
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.