Download Vector Equation of a line (2D)

Vectors (6)
•Vector
Equation of a Line
Revise: Position Vectors
z
A
In 2D and 3D, all points
have position vectors
a
y
e.g. The position vector of point A
 x
 
a   y
z
 
o
x
a = xi + yj + zk
Revise: Parallel Vectors
-20 i + 30 j
-10 i + 15 j
2a
a
-2 i + 3 j
Vectors with a scaler
applied are parallel
1/
5
a
i.e. with a different
magnitude but same
direction
Vector Equation of a line (2D)
y
a
o
A
Any parallel vector (to line)
(any point
it passes
through)
x
A line can be identified by a linear combination
of a position vector and a free vector
Vector Equation of a line (2D)
y
Any parallel vector to line
a
o
A
(any point
it passes
through)
x
A line can be identified by a linear combination
of a position vector and a free vector
Vector Equation of a line (2D)
y
A line can be identified by a linear combination
of a position vector and a free vector
A
b = pi + qj
parallel vector to line
a = xi + yj
x
o
E.g. a + tb
t is a scaler
- it can be any number,
since we only need a parallel vector
= (xi + yj) + t(pi + qj)
Vector Equation of a y = mx + c (1)
1. Position vector to
any point on line
y=x+2
[]
1
3
2. A free vector parallel
to the line
2
2
3. linear combination
[]
of a position vector
and a free vector
[] [] []
x
y
= 1 +t
Equation
3
2
2
Scaler (any number)
[]
[]
2
2
1
3
Vector Equation of a y = mx + c (2)
1. Position vector to
any point on line
y=x+2
[]
4
6
2. A free vector parallel
to the line
-3
-3
[]
3. linear combination
of a position vector
and a free vector
[] [] []
x
y
-3
= 4 +t
Equation
6
[]
-3
-3
-3
Scaler (any number)
[]
4
6
Vector Equation of a y = mx + c (3)
1. Position vector to
any point on line
y = 1/2 x +
3
[]
2
4
2. A free vector parallel
to the line
4
2
[]
3. linear combination
of a position vector
and a free vector
[] [] []
x
y
=
Equation
2
+t
4
4
2
Scaler (any number)
[]
4
2
[]
2
4
Sketch this line and find its equation
[] [] []
= 1 +t
x
y
2
1
3
y = 3x - 1
When t=0
[] []
x
y
=
1
2
x=1, y=2
When t=1
[] []
x
y
=
2
5
x=2, y=5
[]
1
3
[]
1
2
Equations of straight lines
y = 3x - 1
….. is a Cartesian Equation
of a straight line
[ ] =[ ] [ ]
x
y
1
+t
2
1
3
….. is a Vector Equation
of a straight line
Often written …….
r =[
] []
1
+t
2
Any
point
1
3
Direction
r is the position vector of
any point R on the line
Convert this Vector Equation into Cartesian form
r =[
Gradient =
] []
7
+t
3
Increase in y
2
5
the direction vector
Increase in x
Gradient (m) = 5 / 2 = 2.5
[ ] =[ ] [ ]
x
y
7
3 +t
When t = 0
x
7
y = 3
[][]
2
5
x=7
y=3
Equations of form y= mx+c
y= 2.5x + c
3 = 2.5 x 7 + c
c = -14.5
y= 2.5x – 14.5
Convert this Vector Equation into Cartesian form (2)
] []
[ ] =[ ]+ t [ ]
r =[
x
y
Convert to
Parametric
equations
Eliminate ‘t’
subtract
7
+t
3
7
3
x = 7 + 2t
y = 3 + 5t
5x = 35 + 10t
2y = 6 + 10t
5x – 2y = 29
2
5
2
5
Convert this Cartesian equation into a Vector equation
Want something
like this ……….
y = 4x + 3
r =[
] []
a
b +t
Any
point
When x=0, y = 4 x 0 + 3 = 3
[]
0
3
= Any
point
Gradient (m) = 4
[]
1
4
Gradient =
represents
the direction
Increase in y
Increase in x
r =[
1
m
the
direction
vector
= 4
1
] []
0
3 +t
1
4
Convert this Cartesian equation into a Vector equation
Easier Method
y = 4x + 3
Write:
t = 4x
t=y-3
y - 3 = 4x = t
x=
y=
3 +
1/
4
t
t
[ ] =[ ] [ ]
x
y
r =[
0
3 +t
] []
0
3 +t
1
4
1/
4
1
Can replace
with a parallel
vector
Summary
y
A line can be identified
by a linear combination
of a position vector
and a free [direction] vector
r =[
] []
a
b +t
Any
point
1
m
the
direction
vector
(any point it
A passes through)
a
o
Any parallel
vector (to line)
x
Equations of form y-b=m(xa)
Line goes through (a,b) with gradient m