Solution to Exercise 3.8.
Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important… The knowledge doesn't make life any sweeter or happier, does it?
—Helen Keller (1880–1968)

Exercise 3.8. Find the line l[l₁, l₂, l₃] for each problem.

(a)  x₁-axis. 

Two of the points on the x₁-axis are (0, 0, 1) and (1, 0, 1). Hence, by Proposition 3.1, the equation of the line is  Evaluate the determinant to obtain x₂ = 0. Thus the line is [0, 1, 0].

 

(b)  x₂-axis. 

Two of the points on the x₂-axis are (0, 0, 1) and (0, 1, 1). Hence, by Proposition 3.1, the equation of the line is  Evaluate the determinant to obtain x₁ = 0. Thus the line is [1, 0, 0].

 

(c)  The line where all points have the same first and second coordinates.
Two of the points that have the same first and second coordinates are (0, 0, 1) and (1, 1, 1). Hence, by Proposition 3.1, the equation of the line is  Evaluate the determinant to obtain –x₁ + x₂ = 0. Thus the line is [–1, 1, 0] or [1, –1, 0].