Transcript PPT
MAT 1236
Calculus III
Section 12.5 Part I
Equations of Line and
Planes
http://myhome.spu.edu/lauw
HW…
WebAssign 12.5 Part I
Preview
Equations of Lines
Equations of Planes
• Vector Equations
• Parametric Equations
• Symmetric Equations
Recall: Position Vectors
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
In 2D, what kind of info is required to
determine a line?
• Type 1:
• Type 2:
Q: How to extend these ideas?
Vector Equations
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
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
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
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
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
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.