Line and Definition of Slope

      Slope of a line (steepness)
Consider a particle moving along a non vertical line segment from a point p1( x1,y1 ) to a point p1( x1,y1 ). The vertical change y2 – y1 is called the rise, and the horizontal change x2 – x1 the run.


      DEFINITION
If P(x1, y1) and P(x2, y2) are points on a nonvertical line, then slope m of the line is defined by:


It does not matter which point is called P1 and which one is called P2
      Slope of  P1P2

= (y2 - y1)/(x2 - x1)

= -(y1 - y2)/[-(x1 - x2)]

= (y1 - y2)/(x1 - x2) = Slope of P1P2


Any two distinct points on a nonvertical line can be used to calculate the slope of the line. To measure the slope, we generally move from left to right when measuring the distance travelling horizontally.
Because of this, sometimes the concept of fall replaces thar of rise!


      Example
In each part find the slope of the line through
          (A)     (6, 2) and (9, 8)
          (B)     (2, 9) and (4, 3)
          (C)    (-2, 7) and (5, 7)


Solution:
We know that slope of line through two points P1(x1, y1) and p1(x1, y1) , is given by
m= (y2 - y1)/ (x2 - x1)
So
    a) m= (8 - 2)/(9 - 6) = 6/3 = 2
On coordinate plane xy


Similarly
    b) m= (3 - 9)/(4 - 2) = -6/2 = -3
On coordinate plane xy


Also
    c) m= (7 -7)/[5 - (-2)] = 0/7 = 0
On coordinate plane xy


      Definition (Angle of inclination)
For a line L is not parallet to x-axis, the angle of inclination is the smallest angle φ messured counterclockwise from the direction of the positive x-axis to L.
For a line parallel to the x-axis, we take φ = 0
As shown in the following figures.
       


If m is slope of line then,
m = rise/run
    = Rate of change of y with respect to x


      THEOREM
For a nonvertical line, the slope m and the angle of inclination φ are related by
            m = tan φ


      Example:
Find the angle of inclination for a line of slope m = 1 and the angle of inclination for a line of slope m = -1


      Solution:
  If m=1 tan φ = 1, so that φ = π/4 = 45°

  If m=-1 tan φ = -1 since, 0 < φ < π φ = 3π/4 = 135°


      THEOREM Let L1 and L2 be lines with slopes m1 and m2, respectively
  (a)   The lines are parallel if and only if      m1 = m2
  (b)   The lines are prependicular if and only if      m1m2 = -1


      Proof: (a)
If L1 and L2 are non vertical lines, then their angles of inclination φ1 and φ2 are equal.
            φ12
Thus
    m1 = tanφ1 = tanφ2 = m2

Conversely, if two slope lines are equal, I.e.
        M1 = M2
⇒     tan(φ1) = tan(φ2)
⇒         φ1 = φ2
So, lines are parallel.


(b) Assume that φ1 < φ2
Then referring to the figure
m1 = tanφ1 = c/h

m2 = tanφ2 = -h/c


The proof of the converse is left as an exercise.


      THEOREM
The vertical line through (a, 0) and the horizontal line throudgh (0, b) are represented, respectively, by the equation
x = a and y = b
      THEOREM
The line Passing throudh P1(x1, y1) and having slope m is given be the equation
            y - y1 = m(x - x1)
This is called the point-slope form of the line.
      THEOREM
The line with y-intercept b and slope m is given by the equation
            y = mx + b
This is called the slope-intercept form of the line.

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