assigned work: pg. 433 #1-12 equation of a line – slope and point or two points but now we will...
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Assigned work: pg. 433 #1-12
Equation of a line – slope and point or two points
BUT NOW we will learn to describe an Equation of a Line by using vectors…………………..
8.1 Parametric & Vector Equations of Line in a Plane
Vector Equation of a Line in a Plane:
OR
0r r td where t R �������������������������� ��
0 0( , ) ( , ) ( , )x y x y t a b
8.1 Parametric & Vector Equations of Line in a Plane
The vectors and variables from the previous slide are defined below:
0 0 0
( , )
int
( , )
int
( , )
( )
var
r x y is the position vector of
any po on the line
r x y is the position vector of
a particular po on the line
d a b is the direction vector for
the line
t is the parameter which
connects the other two or more
iables i
��������������
��������������
n an indriect manner
8.1 Parametric & Vector Equations of Line in a Plane
Note:
A direction vector can give the magnitude of the slope of a line
Any parallel vector to a line has a direction vector that is the same or scalar multiple of the direction vector in the given line.
( , )d a b��������������
0b
m aa
8.1 Parametric & Vector Equations of Line in a Plane
Ex1:
a)Find the vector equation of the line that passes through points A(3,4) and B(7,2).
b)What is the slope of the line?
0
(4, 2) Re !!!
(2, 1)
(3,4) ( (7,2))
(3,4) (2, 1)
d AB
duce direction vector
r or you can use
r t where t R
����������������������������
��������������
1
2m
8.1 Parametric & Vector Equations of Line in a Plane
c) Determine a different point on the line.
(just choose any value for t …)
When t = 1:
Therefore:
( , ) (3,4) 1(2, 1)
( , ) (5,3)
(5,3) int
x y
x y
is another po on the line
8.1 Parametric & Vector Equations of Line in a Plane
This vector equation can be used to describe a line in terms of its components. An equation written in terms of its components is called the……
“Parametric Equation of a Line”
0
0
x x at where t R
y y bt
8.1 Parametric & Vector Equations of Line in a Plane
Ex2:
A line passes through the point P(-1,1) and has a direction vector
a)State the parametric equation of the line.
1 3
1 2
x t
y t where t R
(3,2)d ��������������
8.1 Parametric & Vector Equations of Line in a Plane
Ex2:
b) Does the point (-4,1) lie on the line?
(*****if it does (-4,1) will produce the same parameter for both components)
Check:
Since parameter is NOT the same for both components, the point DOES NOT lie on the line.
4 1 3 1 1 2
3 3 0 2
1 0
t t
t t
t t
8.1 Parametric & Vector Equations of Line in a Plane
Ex2:
c) Determine the y-intercept of the line.
y int (x=0)
0
0
0 1 3
1
31
1 23
5
3
x
x
t
t
y
y
8.1 Parametric & Vector Equations of Line in a Plane
Parametric equations of a line can be used to find a third form of an equation called
“Symmetric Equation of a Line in a Plane”
It eliminates the parameter from the parametric equations
0 0x x y y
a b
8.1 Parametric & Vector Equations of Line in a Plane
Ex 3:
Find the symmetric equation of the line
Therefore the symmetric equation is:
1 3 7 4
1 7
3 4
x t y t
x yt t
(1,7) ( 3,4)r t
1 7
3 4
x y