lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other, i.e., m = m or, m m = – . Let us consider the following example. Example Find the slope of the lines: (a) Passing through the points ( , – ) and (– , ), (b) Passing through the points ( , – ) and ( , – ), (c) Passing through the points ( , – ) and ( , ), (d) Making inclination of ° with the positive direction of x -axis. Solution (a) The slope of the line through ( , – ) and (– , ) is ( ) m −− = − −− (b) The slope of the line through the points ( , – ) and ( , – ) is – – (– ) = – m (c) The slope of the line through the points ( , – ) and ( , ) is Fig .
MATHEMATICS – (– ) – m , which is not defined. (d) Here inclination of the line α = °. Therefore, slope of the line is m = tan ° = . .
. Angle between two lines When we think about more than one line in a plane, then we find that these lines are either intersecting or parallel. Here we will discuss the angle between two lines in terms of their slopes. Let L and L be two non-vertical lines with slopes m and m , respectively.
If α and α are the inclinations of lines L and L , respectively. Then α tan and α tan m m We know that when two lines intersect each other, they make two pairs of vertically opposite angles such that sum of any two adjacent angles is °. Let θ and φ be the adjacent angles between the lines L and L (Fig10. ).
Then θ = α – α and α , α ≠ °. Therefore tan θ = tan ( α