Angle between 2 lines

Describes how to find the angle between 2 lines

1. Date: Tuesday 10th May 2011
Angle between 2 Lines
Preliminary Extension Mathematics

2. Angle between 2 lines
y
x
0
α β
θ
l
1l
2line l
1
has gradient m
1
line l
2
has gradient m
2
∴m
1
=tanα and m
2
=tanβ

3. Angle between 2 lines
y
x
0
α β
θ
l
1l
2line l
1
has gradient m
1
line l
2
has gradient m
2
∴m
1
=tanα and m
2
=tanβ
and α+θ=β (Why?)

4. Angle between 2 lines
y
x
0
α β
θ
l
1l
2line l
1
has gradient m
1
line l
2
has gradient m
2
∴m
1
=tanα and m
2
=tanβ

and α+θ=β
(Exterior angle of V)

5. Angle between 2 lines
y
x
0
α β
θ
l
1l
2So
θ=β−α

6. Angle between 2 lines
y
x
0
α β
θ
l
1l
2So
θ=β−α
∴tanθ=tan(β−α)

7. Angle between 2 lines
y
x
0
α β
θ
l
1l
2So
θ=β−α
∴tanθ=tan(β−α)
=
tanβ−tanα
1+tanβtanα
You will learn this formula later

8. Angle between 2 lines
y
x
0
α β
θ
l
1l
2So
θ=β−α
∴tanθ=tan(β−α)
=
tanβ−tanα
1+tanβtanα
Why?
=
m
1
−m
2
1+m
1
m
2

9. Angle between 2 lines
y
x
0
α β
θ
l
1l
2So
θ=β−α
∴tanθ=tan(β−α)
=
tanβ−tanα
1+tanβtanα
When tan θ is positive, θ is acute.
When tan θ is negative, θ is obtuse.
=
m
1
−m
2
1+m
1
m
2

10. Angle between 2 lines
y
x
0
α β
θ
l
1l
2
Thus for two lines of gradient
tanθ=
m
1
−m
2
1+m
1
m
2
m
1
and m
2
the acute angle between them is given by
Note that m
1
m
2
≠−1
what does this mean?

11. Angle between 2 lines
y
x
0
α β
θ
l
1l
2
Thus for two lines of gradient
tanθ=
m
1
−m
2
1+m
1
m
2
m
1
and m
2
the acute angle between them is given by
Note that m
1
m
2
≠−1
the formula does not work for perpendicular lines

12. Example 1
Find the acute angle between
y=2x+1
and
y=−3x−2
(to nearest degree)

13. Example 1
Find the acute angle between
y=2x+1∴m
1
=2 and m
2
=−3
and
y=−3x−2
(to nearest degree)

14. Example 1
Find the acute angle between
tanθ=
m
1
−m
2
1+m
1
m
2
y=2x+1∴m
1
=2 and m
2
=−3
and
y=−3x−2
(to nearest degree)

15. Example 1
Find the acute angle between
tanθ=
m
1
−m
2
1+m
1
m
2
y=2x+1∴m
1
=2 and m
2
=−3
and
y=−3x−2
(to nearest degree)
∴tanθ=
2+3
1−6

16. Example 1
Find the acute angle between
tanθ=
m
1
−m
2
1+m
1
m
2
y=2x+1∴m
1
=2 and m
2
=−3
and
y=−3x−2
(to nearest degree)
∴tanθ=
2+3
1−6
∴tanθ=−1

17. Example 1
Find the acute angle between
tanθ=
m
1
−m
2
1+m
1
m
2
y=2x+1∴m
1
=2 and m
2
=−3
and
y=−3x−2
(to nearest degree)
∴tanθ=
2+3
1−6
∴tanθ=−1∴tanθ=1

18. Example 1
Find the acute angle between
tanθ=
m
1
−m
2
1+m
1
m
2
y=2x+1∴m
1
=2 and m
2
=−3
and
y=−3x−2
(to nearest degree)
∴tanθ=
2+3
1−6
∴tanθ=−1∴tanθ=1→θ=45°

19. Example 2
Find the acute angle between
3x−2y+7=0
and
2y+4x−3=0
(to nearest degree)

20. Example 2
Find the acute angle between
3x−2y+7=0∴2y=3x+7
and
2y+4x−3=0
(to nearest degree)

21. Example 2
Find the acute angle between
3x−2y+7=0∴2y=3x+7
and
2y+4x−3=0
(to nearest degree)
∴y=
3
2
x+
7
2

22. Example 2
Find the acute angle between
3x−2y+7=0∴2y=3x+7
and
2y+4x−3=0
(to nearest degree)
∴y=
3
2
x+
7
2
∴m
1
=
3
2

23. Example 2
Find the acute angle between
3x−2y+7=0∴2y=3x+7
and
2y+4x−3=0
(to nearest degree)
∴y=
3
2
x+
7
2
∴m
1
=
3
2
similarly
∴2y=−4x+3
∴y=−2x+
3
2
∴m
2
=−2

24. Example 2
Find the acute angle between
3x−2y+7=0
and
2y+4x−3=0
(to nearest degree)
m
1
=
3
2
applying the formula
m
2
=−2
tanθ=
m
1
−m
2
1+m
1
m
2

25. Example 2
Find the acute angle between
3x−2y+7=0
and
2y+4x−3=0
(to nearest degree)
m
1
=
3
2
applying the formula
m
2
=−2
tanθ=
m
1
−m
2
1+m
1
m
2∴tanθ=
3
2
+2
1−3

26. Example 2
Find the acute angle between
3x−2y+7=0
and
2y+4x−3=0
(to nearest degree)
m
1
=
3
2
applying the formula
m
2
=−2
tanθ=
m
1
−m
2
1+m
1
m
2∴tanθ=
3
2
+2
1−3
∴tanθ=
−7
4

27. Example 2
Find the acute angle between
3x−2y+7=0
and
2y+4x−3=0
(to nearest degree)
m
1
=
3
2
applying the formula
m
2
=−2
tanθ=
m
1
−m
2
1+m
1
m
2∴tanθ=
3
2
+2
1−3
∴tanθ=
−7
4
∴tanθ=
7
4
→θ=60°

28. Example 3 - by thinking and
drawing....
Find the acute angle between
y=x+3
and
y=−3x+5
(to nearest degree)

29. Example 3 - by thinking and
drawing....
Find the acute angle between
y=x+3
and
y=−3x+5
(to nearest degree)
y
x
0
α β
θ
y=x+3y=−3x+5

30. Example 3 - by thinking and
drawing....
Find the acute angle between
y=x+3
and
y=−3x+5
(to nearest degree)
y
x
0
α β
θ
m
1
=1y=x+3y=−3x+5→α=45°

31. Example 3 - by thinking and
drawing....
Find the acute angle between
y=x+3
and
y=−3x+5
(to nearest degree)
y
x
0
α β
θ
m
1
=1m
2
=−3y=x+3y=−3x+5→α=45°→β=108°

32. Example 3 - by thinking and
drawing....
Find the acute angle between
y=x+3
and
y=−3x+5
(to nearest degree)
y
x
0
α β
θ
m
1
=1m
2
=−3y=x+3y=−3x+5→α=45°→β=108°But α+θ=β

33. Example 3 - by thinking and
drawing....
Find the acute angle between
y=x+3
and
y=−3x+5
(to nearest degree)
y
x
0
α β
θ
m
1
=1m
2
=−3y=x+3y=−3x+5→α=45°→β=108°But α+θ=β∴θ=63°