Angle between two lines

Jun 18, 2020, 16:45 IST

Angle between two lines

Definition :-When two lines intersect, the angle between them is defined as the angle through which one of the lines must be rotated to make it coincide with the other line.

Formula description :-
First, notice that when two lines intersect, one of the two pairs is acute and the other pair is obtuse. The angle between two lines is defined as the smallest of these angles or the acute angle denoted by θ.

We are going to use the inclinations of the two lines to find the angle between the two lines. You may need to review the lesson about inclination of line.

Now, you can use the formula for the tangent of the difference of two angles.

Third, in the lesson about inclination of a line, we learned that

After substituting m1 for tan θ1 and m2 for tan θ2 in the equation immediately above, we get:
Notice the use of the absolute value to ensure that tan θ is equal to a positive number.

Example 1 :-Find the angle between the following two lines.
Line 1: 3x -2y = 4
line 2: x + 4y = 1

Solution :-Put 3x - 2y = 4 into slope-intercept form so you can clearly identify the slope.
3x - 2y = 4
2y = 3x – 4
y = 3x / 2 - 4/2
y = (3/2)x – 2

put x + 4y = 1 into slope-intercept form so you can clearly identify the slope.
X + 4y = 1
4y = -x + 1
y = -x/4 + 1/4
Y = (-1/4)x + 1/4

the slopes are 3/2 and -1/4 or 1.5 and -0.25. It does not matter which one is m1 or m2. You will get the same answer.
Let m1 = 1.5 and m2 = -0.25

Example 2 :-Find the equation of line through point (3,2) and making angle 45° with the line x-2y = 3

Solution :-
Let m be the slope of the required line passing through (3,2). So, using slope point form, its equation is
Y-2 = m(x-3)--------- (i)
Slope of line x-2y = 3 is 1/2.
Since, these lines make an angle of 45° so,
Substituting values of m in equation (i), we get
Y-2 = 3(x-3) and y-2 = -(x-3)/3
Or, 3x-y-7 = 0 and x+3y-9 = 0 are the required equations of line.