Show that lines (x + 1)/-10 = (y + 3)/-1 = (z - 4)/1 and (x + 10)/-1 = (y + 1)/-3 = (z - 1)/4 intersect each other.

Question

Show that lines \(\frac{x + 1}{-10} = \frac{y + 3}{-1} = \frac{z - 4}{1}\) and \(\frac{x + 10}{-1} = \frac{y + 1}{-3} = \frac{z - 1}{4}\) intersect each other. Find the co-ordinates of their point of intersection.

(x + 1)/-10 = (y + 3)/-1 = (z - 4)/1 and (x + 10)/-1 = (y + 1)/-3 = (z - 1)/4

Answer 1

The equations of the lines are

From (1), x = -1 -10λ, y = -3 – 2, z = 4 + λ

∴ the coordinates of any point on the line (1) are

(-1 – 10λ, – 3 – λ, 4 + λ) 

From (2), x = -10 – u, y = -1 – 3u, z = 1 + 4u 

∴ the coordinates of any point on the line (2) are

(-10 – u, -1 – 3u, 1 + 4u) 

Lines (1) and (2) intersect, if 

(- 1 – 10λ, – 3 – λ, 4 + 2) = (- 10 – u, -1 – 3u, 1 + 4u) 

∴ the equations -1 – 10λ = -10 – u, -3 – 2= – 1 – 3u 

and 4 + λ = 1 + 4u are simultaneously true. 

Solving the first two equations, we get, λ = 1 and u = 1. These values of λ and u satisfy the third equation also.

∴ the lines intersect.

Putting λ = 1 in (-1 – 10λ, -3 – 2, 4 + 2) or u = 1 in (-10 – u, -1 – 3u, 1 + 4u), we get 

the point of intersection (-11, -4, 5).