Question

Show that the lines given by `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1) and (x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` intersect. Also, find the coordinates of their point of intersection.

Answer 1

The equations of the lines are 

`(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` = λ                ...(say)...(1)

and `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` = μ        ...(say)...(2)

From (1), x = – 1 – 10λ, y = – 3 – λ , z = 4 + λ 
∴ the coordinates of any point on the line (1) are  (–1 – 10λ, – 3 – λ, 4 + λ)

From (2), x = – 10 – μ, y = – 1 – 3μ, z = 1 + 4μ
∴ the coordinates of any point on the line (2) are  ( – 10 – μ, – 1 – 3μ, 1 + 4μ)

Lines (1) and (2) intersect, if (– 1 – 10λ, – 3 – λ, 4 + λ) = ( – 10 – μ, – 1 – 3μ, 1 + 4μ)
∴ the equation – 1 – 10λ = – 10 – μ, – 3 – λ = –1 – 3μ and 4 + λ = 1 + 4μ are simultaneously true.

Solving the first two equations, we get, λ = 1, and μ = 1.
These values of λ and μ satisfy the third equation also.
∴ the lines intersect.
Putting λ = 1 in (– 1 – 10λ, – 3 – λ, 4 + λ) or  μ = 1 in  (– 10 – μ, –1 – 3μ, 1 + 4μ), we get the point of intersection (–11, – 4, 5).