What is a Slope (Gradient)?
The slope of a line is a number that measures its "steepness". It is the change in y for a unit change in x along the line.
The slope of a line is generally denoted by m.
m = tan θ
Since a line parallel to x – axis makes an angle of 0° with x – axis, therefore its slope is tan 0° = 0.
A line parallel to y – axis, perpendicular to x – axis makes an angle of 90 with x – axis, so its slope is tan 90° = ∞. Also, the slope of a line equally inclined with axes is 1 or -1 as it makes 45° or 135° with x – axis.
The angle of inclination of a line with the positive direction of x – axis in anticlockwise sense always lies between 0° and 180°.
Slope of a line in terms of coordinates of any two points on it
Let P (x1, y1) and Q (x2, y2) be two points on a line making an angle θ with the positive direction of x – axis. Draw PM, QN perpendiculars on x – axis and PL perpendicular on QN.
Then, PL = MN = ON – OM = x2 – x1
and QL = ON – LN = QN – PM = y2 – y1
In ΔPQL, tan θ = QL/PL = (y2 – y1)/(x2 – x1)
Thus, if (x1, y1) and (x2, y2) are coordinates of any two points on a line, then its slope is
m = (y2 – y1)/(x2 – x1)
m = Difference of ordinates
Difference of abscissae
or, m = Vertical Step
Angle between two lines
The angle θ between the lines having slopes m1 and m2 is given by
tan θ = + m2 – m1
1 + m1m2
Slope (Gradient) of Parallel lines
If two lines of slopes m1 and m2 are parallel, then the angle θ between them is 0°.
tan θ = tan 0° = 0
m2 – m1 = 0
1 + m1m2
m2 = m1
Thus, when two lines are parallel, their slopes are equal.
Slope (Gradient) of Perpendicular lines
If two lines of slope m1 and m2 are perpendicular, then the angle θ between them is of 90°
cot θ = 1 + m1m2
m2 – m1
0 = 1 + m1m2
m1m2 = -1
Thus, when two lines are perpendicular, the product of their slopes is -1. If m is the slope of a line, then the slope of a line perpendicular to it is - (1/m).
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