Write the equation 4x + 5y – 3 = 0 in normal form. Then, find the length of the normal and the angle it makes with the positive x–axis.

Answer 1

From
4x + 5y – 3 = 0
Divide all coefficients by sqrt(4^2+5^2) to get
4x/sqrt(41) +5y/sqrt(41) - 3/sqrt(41) = 0
where
3/sqrt(41) = length of normal
cos(θ)=4/sqrt(41), and
sin(θ)=5/sqrt(41)

Note: the normal form is given by:
x cos(θ) + y sin(θ) - p =0
where |p|=length of normal.