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13 B• Lines in 2D and
3D
The vector AB and the vector equation of the line AB are very different things.
A
B
x
xB
Ax
x
The line AB is a line passing through the points A and B and has infinite length.
The vector AB
The line through A and B.
The vector AB has a definite length ( magnitude ).
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Vector Equation of a Line
Finding the Equation of a LineIn coordinate geometry, the equation of a line is
e.g. 32 xy
The equation gives the value (coordinate) of y for any point which lies on the line.
The vector equation of a line must give us the position vector of any point on the line.We start with fixing a line in space.We can do this by fixing 2 points, A and B. There is only one line passing through these points.
y mx b
Vector Equation of a Line
In Other Words:
We can determine the vector equation of a line using its direction and any fixed point on the line.
x
B
A
x
Ox
We consider several more points on the line.
2Rx
3Rx
1Rx
A and B are fixed points.
We need an equation for r, the position vector of any point R on the line.
r1a
Starting with R1: 1r aAB2
1
Vector Equation of a Line
x
B
A
x
Ox
B
AWe consider several more points on the line.
x
1R
2R
3R
x
x
xa
r2
A and B are fixed points.
Starting with R1:
2r
1r aAB2
1
a AB2
We need an equation for r, the position vector of any point R on the line.
Vector Equation of a Line
x
B
A
x
Ox
B
AWe consider several more points on the line.
Ox
1R
2R
3R
x
x
xa
r3
A and B are fixed points.
Starting with R1:
2r
1r aAB2
1
a AB2
3r a AB)( 41
We need an equation for r, the position vector of any point R on the line.
Vector Equation of a Line
x
B
A
x
Ox
B
A
x
1R
2R
3R
x
x
xa
O
So for R1, R2 and R3
For any position of R, we have
t is called a parameter and can have any real value.It is a scalar not a vector.
Vector Equation of a Line
1
2
3
1
2
2
1
4
r a AB
r a AB
r a AB
��������������
��������������
��������������
r a ABt ��������������
x
B
A
x
Ox
B
A
x 1Rx
x
xa
O
This is what I’m Saying:
t is called a parameter and can have any real value.
It is a scalar not a vector.
Fixed Point
Fixed Point
Any Poin
t
b
Say you have a line that passes through a fixed point A with position vector a, and that line is parallel to vector b. Point R is on the line
OR r��������������
r
By vector addition:OR OA AR ������������������������������������������
Vector Equation of a Line
r a ABt ��������������
x
B
A
x
Ox
B
A
x 1Rx
x
xa
O
Fixed Point
Fixed Point
Any Poin
t
b
Since is parallel to b,
r
AR tb�������������� AR��������������
r a tb
Vector Equation of a Line
direction vector . . .
SUMMARY
The vector equation of the line through 2 fixed points A and B is given by
The vector equation of the line through 1 fixed point A and parallel to the vector is given by
p
ptar
position vector . . .of a known point on the line
of the line
r a t AB r a tb ��������������
Vector Equation of a Line
In 2-D, we are dealing with a line in a plane
A
(a1 , a2
)
R
(x , y)
1 1
2 2
a bxt
a by
The vector equation of the line
1
2
bb
b
Is the direction vector
1
2
bb
b
Slope or gradient
is:2
1
bm
b
We can write the parametric equations of the line:
1 1
2 2
a b tx
a b ty
1 1 2 2&x a b t y a b t Each point on the line corresponds
to exactly one value of t.Convert into Cartesian form by equating t
values.1 2
1 2
x a y at
b b
We get the Cartesian equation of
the line as: 2 1 2 1 1 2b x b y b a b a
A line passes through the point A(1,5) and has the direction vector . Describe the line using:3
2
Draw a sketch
A
(1 , 5)
R
(x , y)
3
2b
O
ra
1 3&
5 2a OA b
��������������
A) A vector equation
1 3
5 2
xt
y
A line passes through the point A(1,5) and has the direction vector . Describe the line using:3
2
A
(1 , 5)
R
(x , y)
3
2b
O
ra
B) Parametric Equations
1 3 & 5 2x t y t
1 1
2 2
a b tx
a b ty
A line passes through the point A(1,5) and has the direction vector . Describe the line using:3
2
A
(1 , 5)
R
(x , y)
3
2b
O
ra
C) Cartesian Equation
1 2
1 2
x a y at
b b
1 5
3 2
x yt
2 1 2 1 1 2b x b y b a b a 2 2 3 15
2 3 13
x y
x y
In 3-D, we are dealing with a line in Space
A
(a 1,a2,a3
)
R
(x,y,z)
1 1
2 2
3 3
a bx
a bty
a bz
Is the vector equation of the line
1
2
3
b
bb
b
Is the direction vector
1
2
3
b
bb
b
Slope or gradient NONE. We just describe its direction by its direction vector
We can write the parametric equations of the line:
1 1
2 2
3 3
a b tx
a b ty
a b tz
Each point on the line corresponds to exactly one value of t.
Find a vector equation and the corresponding parametric equations of the line through (1,-2, 3) in the direction of 4i + 5j – 6k
The parametric equations are:
1 4
2 5
3 6
x
ty
z
The vector equation is: r = a + tb
1 1
2 2
3 3
a b tx
a b ty
a b tz
1 4
2 5
3 6
x t
y t
z t
Non-Uniqueness of a line
Consider the line passing through (5,4) and (7,3). When writing the equation of the line, we could use either point to give the position vector a
(5,4)(7,3)
We could also use theDirection vector
2
1
2
1
We could also use the Direction vector
2
1
We could also use ANY Direction vector that is a non-zero scalar multiple of these vectors
Non-Uniqueness of a line
So, we could write the equation of the line as:
(5,4)(7,3)
2
1
5 2
4 1x t
7 2
3 1x s
We could keep going.
Non-Uniqueness of a lineSo, we could write the equation of the line as:
(5,4)(7,3)2
1
5 2
4 1x t
7 2
3 1x s
Non-Uniqueness of a lineFind a parametric equations of the line through A(2,-1,4) and B(-1,0,2)
We need a direction vector for the line, either
AB or BA����������������������������
1 2 3
0 1 1
2 4 2
AB
��������������Using point A, the equations are: 2 3
1
4 2
x t
y t
z t
Using point B, the equations are: 1 3
2 2
x s
y s
z s
Homework Page 325 (1 –
8)