The Vector Equation of a Line

The Angle Between 2 Lines13C

The Angle Between 2 LinesThe angle between 2 vectors is in the range0 ≤ θ ≤ 180°

b1

b2

And we used the formula:

1 2

1 2

b bCosb b

The Angle Between 2 LinesWhen the lines go infinitely, we usually refer to the acute angle. So, Cosθ > 0

b1 b2

And we used the formula:

1 2

1 2

b bCos

b b

b1 and b2 are the direction vectors

L1

L2

The Angle Between 2 LinesFind the angle between the lines

1

2

: 2 3 , 1: 1 2 , 4 3

L x t y tL x s y s

1 2

3 2,

1 3b b

What do these represent?

1 2

1 2

b bCos

b b

6 30.2631

10 13Cos

74.7

The Angle Between 2 LinesFind the angle between the lines

1 1 2 2

2 3 1 3: r 1 1 : r 0 1

4 2 2 2L s L t

What do you notice?

1 2? ?b b

Since b1 = b2, the lines are parallel and the angle between them is 0°

The Angle Between 2 LinesFind the ACUTE angle between the lines

1 2: 2 5 : 3x-2y=8L x y L 22 5 has a gradient of - 1

1Has a direction vector (Call it "a")

2

x y

33 2 8 has a gradient of 2

2Has a direction vector (Call it "b")

3

x y

If a line has a gradient

of

ba

Then it has a direction vector of

ab

The Angle Between 2 Lines22 5 has a gradient of - 1

1Has a direction vector (Call it "a")

2

x y

33 2 8 has a gradient of 2

2Has a direction vector (Call it "b")

3

x y

(1 2) ( 2 3) 41 4 4 9 5 13

Cos

a bCos

a b

1 4 60.35 13

Cos

The Angle Between 2 Lines

HomeworkPage 327 (1 – 5)

The Angle Between 2 Lines

1 2

12 3 has direction vector and has direction vector

5 4L L

The Angle Between 2 Lines

1 2

12 3 has direction vector and has direction vector

5 4L L

If is the angle betwen them, then cos ?

12 35 4

cos144 25 9 16

The Angle Between 2 Lines

12 35 4

cos144 25 9 16

36 20 16 cos

(13)(5) 65

1 16cos 75.765