Solution to Exercise 3.15.
A geometer went to the beach to catch the rays and became a TanGent.

Exercise 3.15. (a)  What is the relationship between the coordinates of two distinct parallel lines? Justify the expressions. 
Two distinct lines l[l₁, l₂, l₃] and m[m₁, m₂, m₃] are parallel if and only if there is a nonzero constant k such that m₁ = kl₁, m₂ = kl₂, but m₃ does not equal kl₃.

 

Proof. Lines l[l₁, l₂, l₃] and m[m₁, m₂, m₃] are two distinct parallel lines if and only if the system of equations

l₁x₁ + l₂x₂ + l₃ = 0

m₁x₁ + m₂x₂ + m₃ = 0

does not have a unique solution. The system does not have a unique solution if and only if the matrix

 

is singular. The matrix A is singular if and only if the rows of A are linearly dependent. The rows of A are linearly dependent if and only if there exist constants k₁ and k₂ not both zero such that k₁l_i + k₂m_i = 0 for i = 1, 2. Suppose k₂ = 0, then k₁ is nonzero and k₁l_i = 0 for i = 1, 2. Hence, l₁ = l₂ = 0, which contradicts that the first two coordinates of line l cannot both be zero. Thus, k₂ cannot be 0. Hence,  for i =1, 2. Set  Since [l₁, l₂, l₃] ~ [kl₁, kl₂, kl₃] = [m₁, m₂, kl₃] and lines l and m are distinct,  //

(b) Based on the definition of the measure of an angle between two lines, what is the measure of the angle between two parallel lines?  

Two lines p[p₁, p₂, p₃] and q[q₁, q₂, q₃] are parallel if there is a nonzero constant k such that q₁ = kp₁ and q₂ = kp₂. We compute from the definition of the measure of the angle between two lines to obtain