find angle between two lines

Question

Find the angle between two lines

x-y=0

3x-2y+1=0

Answer 1

Write the line equation in slope-intercept form y = mx + b, m is slope and b is y-intercept. x-y=0 x -y +y = 0+y                                  (Add y to each side) y = x                                             Compared to the slope intercept equation y = mx +b slope(m1) = 1, and intercept(b1) = 0 And again the next equation Write the line equation in slope-intercept form y = mx + b, m is slope and b is y-intercept. 3x-2y+1=0 3x -2y +2y+1 = 2y                           (Add to 2y each side) 3x+1 = 2y  2y = 3x +1                                       y = (3/2)x +1/2                                (Divide each side by 2)                                            Compared to the slope intercept equation y = mx +b slope(m2) = 3/2, and intercept(b2) = 1/2 The Angle between two lines  tan(θ) = (m1 -m2)/(1+m1m2) Substitute the value of m1 & m2 above equation tan( θ) = (1 - (3/2))/(1+1(3/2))              = ((-1/2)/(1+3/2)                  (LCM of 1 -3/2 = -1/2)                                 = ((-1/2)/(5/2))                      (LCM of 1 +(3/2) = 5/2)             = - (1/5)                              (Simplify) The angle between lines is (θ) = tan-1 (-(1/5) ) = -11.03O .

 

Comments

tan(θ) = | (m₁ -m₂)/(1+m₁m₂) |.

The angle between lines (θ) = tan^-1 (1/5 ) = tan^-1 (0.2) = 11.309^O .