The equations of the straight line
A
P ( straight line x
0
, y
0
)
in the plane can be defined by
and a direction vector v
 v x
, v y
 one of its points
(there is only one line passing through a point with a certain slope).
Each point Q ( x , y )

PQ is parallel to v
in the straight line has the following property:
Conversely, if a point Q ( x , y ) satisfies that PQ is parallel to v

, then it is in the straight line.
So we have that: Q ( x , y ) is in the line


PQ || v
PQ
   v


for some
  


 x y

 x
0 y
0


 x y






 v x
 v y


 x
0 y
0




for some


 v x v y



for some
  

   the parametric equations of the straight line
Q ( x , y ) = P ( x
0
, y
0
) +
 ·
 v

for some
  
the vector equation of the straight line
The equation of the straight line that passes through point P ( x
0
, y
0
) in the direction of vector
 v
 v x
, v y
 can also have the following different forms:


 x y

 x
0 y
0



 
 v x v y



for some
   


 x
 v y x
0 y v
 x y
0 x
 x
0 x v x
 x
0 v x




 

for some
    x
 x
0 v x
 y
 y
0 v y

 y y
 v y
 v y
Ax
 y y
0
0
By
 v y v x
 v y

C
 x y

 x
 x
0
  v x
 y

0
 x
0

 y
0

 y
0 y
 y
0
  v y x
 m  v
 x
 x
0

the point-slope form
 v y x
0
 v x y
 v x y
0
 v y x
 v x y
 v x y
0
 v y x
0

0

A
 v y
, B
 being
 v x
, C
 v x y
0
 v y x
0
the general form or else
Ax

By

C
A
 v y
, B
  being v x
, C
 v y x
0
 v x y
0
the standard form y
 y
0
 m  v
 x
 x
0

 y
 m  v x
 m  v x
0
 y
 mx
 n y
0
 n
  m  v x being
0
 y
0
, m
 m  v the slope-intercept form

Example: the forms of equation of the straight line that passes through P ( 4 ,

1 ) in the direction of
 v


2 , 3

( x , y )


4 ,

1
 

2 , 3

,
  
the vector equation x y


4


1




 

3


,
   y x


4

1


2

3



,
  
the parametric equations x


4
 y
2

3 x


2
4
 y

1
3
3

2
 x

4

 y

1 y

1


3  x

4
 the point-slope equation
2
3
 x

4

 
2
 y

1

3 x

12
 
2 y

2 3 x

2 y

10

0 the general equation
3 x

2 y

10 the standard equation y

1


3  x

4

2 y


2
3  

1 y


2
3 x

12
2

1 y


2
3 x

5 the slope-intercept equation
2x+y = 5 a linear function y = –2x+5 x | y
1 | 3
2 | 1
3 |–1
4 |–3
Special cases of straight lines

Horizontal line
An equation like y=c is satisfied by the coordinates of all the points in the horizontal line passing through (0,c)
Vertical line
An equation like x=c is satisfied by the coordinates of all the points in the vertical line passing through (c,0)
Main diagonals
The equation y= x describes a straight line through the origin (0,0) that bisects the first and third quadrant.
The equation y= –x describes a straight line through the origin (0,0) that bisects the second and fourth quadrant.
Lines through the origin
An equation like y=mx describes a straight line through the origin (0,0)
Slope or gradient of a straight line
The slope or gradient of a line is that of its direction vector: m
 m  v
 v v x y
When we have the general equation or the equation in standard form the slope is: m
 
A
B
The gradient of a line joining the points P ( x
1
, y
1
) and Q ( x
2
, y
2
) is: m
 y x
2
Straight lines that are parallel must have the same gradient.
When two straight lines are perpendicular , the product of their gradients is –1.
2

 y x
1
1
Note: given the equation in general or standard form, the vector
 u

A , B

is perpendicular to the line.