MAT 1236
Calculus III
Section 12.5 Part I
Equations of Line and
Planes
http://myhome.spu.edu/lauw
HW…
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WebAssign 12.5 Part I
Preview
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Equations of Lines
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Equations of Planes
• Vector Equations
• Parametric Equations
• Symmetric Equations
Recall: Position Vectors
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Given any point P  a1, a2  , OP  a1, a2 is the
position vector of P.
To serve as a position vector, the initial
point O of the vector is fixed.
Equations of Lines
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In 2D, what kind of info is required to
determine a line?
• Type 1:
• Type 2:
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Q: How to extend these ideas?
Vector Equations
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Ingredients
•
•
A (fixed) point P  x , y , z  on the line
A (fixed) vector v=<a,b,c> parallel to the line
0
0
0
0
Any vector parallel to the line can be
represented by ________________
The position vector of a (general) point P  x, y, z 
on the line can be represented by
________________
Parametric Equations
r  r0  tv
x, y, z  x0 , y0 , z0  t a, b, c
v  a, b, c
Example 1
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Find a vector equation and parametric equations for
the line that passes through the point (1,1,5) and is
parallel to the vector <1,2,1>.
Vector Equation
r  r0  tv
Example 1
Vector Equation
r  r0  tv
Example 1: Parametric Equation
x  1  t , y  1  2t , z  5  t
Can you recover (1,1,5) and <1,2,1> from
the parametric equation?
Remarks
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As usual, parametric equations are not
unique (e.g. v1=<-2,-4,-2> gives another
parametric equation.)
Example 1: Symmetric Equation
x  1  t , y  1  2t , z  5  t
Example 1: Symmetric Equation
x  1  t , y  1  2t , z  5  t
Can you recover (1,1,5) and <1,2,1> from
the symmetric equation?
What if…
x  1  t , y  1  2t , z  5  t
If one of the component is a constant,
then…
3 Possible Scenarios
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Given 2 lines in 3D, they are either
•
•
•
Example 2
Show that the 2 lines are parallel.
 L1 : x  1  t , y  1  2t , z  5  t

 L2 : x  5  2s, y  3  4s, z  2s
Example 3
Find the intersection point of the 2 lines
 L1 : x  2t , y  3  4t , z  1  t

 L2 : x  1  s, y  3s, z   s
(The lines intersect if there is a pair of
parameters (s,t) that gives the same point
on the two lines.)
s  1, t  0
 0,3,1
Expectations
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You are expected to carefully explain
your solutions. Answers alone are not
sufficient for quizzes or exams.
Example 4
Show that the two lines are skew.
 L1 : x  1  t , y  2  3t , z  4  t

 L2 : x  2s, y  3  s, z  3  4s
Example 4
Show that the two lines are skew.
 L1 : x  1  t , y  2  3t , z  4  t

 L2 : x  2s, y  3  s, z  3  4s
1. Show that the two lines are not parallel.
2. Show that the two have no intersection
points.
13i  6 j  5k
(1),(2)  t 
11
8
,s 
5
5
Expectations
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To show that two lines are non-parallel,
you are expected to show that the cross
product of the two (direction) vectors is a
non-zero vector.
Do not substitute s and t directly into the
3rd equation. You are expected to
compute the values of the two sides
separately and compare the values.