Slope, Angle of Inclination and Angle Between Two Lines - Analytic Geometry

* Slope of a Line

Formula:

 
(Slope Intercept Form)

y = mx + b

m = rise/run = (y_{2 -} y1) / (x_{2 -} x1)

Where:

x1 and y1 = coordinates of first point

x2 and y2 = coordinates of second point

Note:
- In solving for the slope of a line, there must be first and second (points needed) since there are uncountable points that forms a line. Terms such as "first" and "second" points are just used to avoid confusion because we have to be consistent on what values to be substituted.

Example:

Given:

Two points = (2, 4) and (5, 6)

Required: m

Solution:

First let what to be the first and second point. Whether we let (2, 4) or (5, 6) as our first point, it doesn't matter because it will not affect the answer.

Let:

P1(2,4)
P2(5,6)

m = (y_{2 -} y1) / (x_{2 -} x1)

m = (6 - 4) / (5 - 2)

m = 2/3

(Solution 2)

Let:

P1(5,6)
P2(2,4)

m = m = (y_{2 -} y1) / (x_{2 -} x1)

m = (4 - 6) / (2 - 5)

m = -2 / -3

m = 2/3

- We just got the same answer!

* Angle of Inclination

- It is the angle formed by a line with respect to the positive x-axis.

Formula:

tanϴ = m

Where:

ϴ = angle of inclination
m = slope of the line

Examples:

* Determine the angle of inclination of a line passing through (-3, -2) and (5, 1).

Given: 

P1(-3, -2)
P2(5, 1)

Required: ϴ

Solution:

tanϴ = m

ϴ = tan^-1 [(y_{2 -} y1) / (x_{2 -} x1)]

ϴ = tan^-1 [(1 - (-2)) / (5 - (-3))] 

ϴ = tan^-1 [(1 + 2) / (5 + 3)] 

ϴ = tan^-1(3/8)

ϴ = 20.36 degrees

* Find the slope of the line having an angle of inclination equal to 45 degrees.

Given:

ϴ = 45 

Required: m

Solution:

tanϴ = m

m = tan(45 degrees)

m = 1

* The equation of the line is 3x - y + 4 = 0. Find its angle of inclination.

Given:

3x - y + 4 = 0

Required: ϴ

Solution:

Make in the form:

y = mx + b

3x - y + 4 = 0

y = 3x + 4

tanϴ = m

ϴ = tan^-1(3)

ϴ = 71.57 degrees

* Angle between Two Lines

    - angle formed by two intersecting lines.

Formula:

ø = ϴ₂ - ϴ₁

tanø = (tanϴ2 - tanϴ₁) / (1 + tanϴ₁tanϴ2)

tanø = (m₂ - m1) / (1 + m1m₂)


Where:

ϴ₁ = angle of the first line
ϴ ₂ = angle of the second line

m1 = slope of the first line
m2 = slope of the second line
ø = angle between the two lines

Note:

    - If you get a negative angle just get its absolute. Perhaps, m₂ and m₁ have just been interchanged.

Example:

* Two intersecting lines have slopes of m = 2 and m = 3. Find the angle between them.

Given:

Let:

m₁ = 2
m₂ = 3

Required: ø

Solution:

tanø = (m₂ - m1) / (1 + m1m₂)

tanø = (3 - 2) / (1 + (2)(3)) 

ø = tan^-1(1/7)

ø = 8.13 degrees