mat 1236 calculus iii section 12.5 part i equations of line and planes
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MAT 1236Calculus III
Section 12.5 Part I
Equations of Line and Planes
http://myhome.spu.edu/lauw
HW…
WebAssign 12.5 Part I
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Equations of Lines• Vector Equations
• Parametric Equations
• Symmetric Equations
Equations of Planes
Recall: Position Vectors
Given any point , is the position vector of P.
To serve as a position vector, the initial point O of the vector is fixed.
1 2,OP a a�������������� 1 2,P a a
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 on the line
• A (fixed) vector v=<a,b,c> parallel to the line
Any vector parallel to the line can be represented by ________________
The position vector of a (general) point on the line can be represented by ________________
0 0 0 0, ,P x y z
, ,P x y z
Parametric Equations
0
0 0 0, , , , , ,
r r tv
x y z x y z 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>.
0
Vector Equation
r r tv
Example 1
0
Vector Equation
r r tv
Example 1: Parametric Equation
Can you recover (1,1,5) and <1,2,1> from the parametric equation?
1 , 1 2 , 5x t y t z t
Remarks
As usual, parametric equations are not unique (e.g. v1=<-2,-4,-2> gives another parametric equation.)
Example 1: Symmetric Equation
1 , 1 2 , 5x t y t z t
Example 1: Symmetric Equation
Can you recover (1,1,5) and <1,2,1> from the symmetric equation?
1 , 1 2 , 5x t y t z t
What if…
1 , 1 2 , 5x t y t z 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.
1
2
: 1 , 1 2 , 5
: 5 2 , 3 4 , 2
L x t y t z t
L x s y s z s
Example 3
Find the intersection point of the 2 lines
(The lines intersect if there is a pair of parameters (s,t) that gives the same point on the two lines.)
1
2
: 2 , 3 4 , 1
: 1 , 3 ,
L x t y t z t
L x s y s z s
1, 0
0,3,1
s t
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.
1
2
: 1 , 2 3 , 4
: 2 , 3 , 3 4
L x t y t z t
L x s y s z s
Example 4
Show that the two lines are skew.
1
2
: 1 , 2 3 , 4
: 2 , 3 , 3 4
L x t y t z t
L x s y s z s
1. Show that the two lines are not parallel.
2. Show that the two have no intersection points.
13 6 5
11 8(1), (2) ,
5 5
i j k
t s
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.
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