You need to use the following formula to evaluate the angle between lines, such that:
`alpha = tan^(-1)|(f_1'(x) - f'_2(x))/(1 + f_1'(x)*f_2'(x))|`
You need to evaluate `f_1'(x)` and `f_2'(x)` such that:
`f_1'(x) = (3x + 1)' => f_1'(x) = 3`
`f_2'(x) = (-2x - 7)' => f_2'(x) = -2`
`alpha = tan^(-1)|(3 - (-2))/(1 - 6)| => alpha = tan^(-1)|-1|`
`alpha = -tan^(-1) 1 => alpha = -pi/4 => alpha = pi - pi/4 => alpha = 3pi/4`
Hence, evaluating the angle within the given lines yields `alpha = 3pi/4.`
We'll note the angle between the lines as a.
We'll apply the formula:
tan a = |(m1 - m2)/(1+m1*m2)| (1)
m1 and m2 are the slopes of the given lines.
We'll note the lines:
d1:y=3x+1 => m1 = 3
d2:y=-2x-7 => m2 = -2
We'll substitute m1 and m2 into the formula (1):
tan a = |(3+2)/(1-6)|
tan a = |5/-5|
tan a = |-1|
tan a = 1
a = arctan 1 + k*pi
a = pi/4 + k*pi
The angle between lines is of 45 degrees.