# Relative position of two lines

Two straight lines,* r* and *s*, can be:

- **Intersecting lines**, if they intersect at a point. If the angle they form is 90^{0}, they are **perpendicular**.

- **Coincident lines**, if they contain the same points.

- **Parallel lines**, if there aren’t any intersection points.

**Case 1**: we know a point and a direction vector of each straight line: *r {A,u}; s {B,v}*

-If *u* and *v* are linearly dependent:

· If* u* and *AB* are linearly dependent, *r* and *s* are coincident lines.

· If *u* and *AB* are linearly independent, *r* and *s* are parallel lines.

- If *u* and *v* are linearly independent: *r* and *s* are intersecting lines.

Example:

**Case 2**: we know a point and the slope of each straight line: *r {A,m _{r}}; s {B,m_{s}}*

- If *m*_{r} =* m*_{s}:

· If *A Є s,* *r* and *s* are coincident lines.

· If *A ¢ s, r* and *s* are parallel lines.

- If *m _{r} ≠ m_{s}*:

*r*and

*s*are intersecting lines.

Example:

**Case 3**: we know the implicit equations of the lines: *r :Ax + By + C = 0; s:A’x+ B’y+ C’ = 0*

Example:

**Exercise**: Determine the relative position of these pairs of straight lines:

a) 3x + 3y - 5 = 0; 6x + 6y -11 = 0

b) y = 3x + 2 ; y - 3 = 2·(x+1)

c) x = 1 + λ

y = 3 -3λ , λ€R ; y = 3x + 2

d) 3x - 3y - 15 = 0; y + 2 = 1·(x - 3)

Solutions: a) parallel; b) intersecting; c) intersecting; d) coincident

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