Question

Show that the lines `(x - 1)/1 = (y - 2)/2 = (z + 1)/-1` and `x/2 = (y - 3)/2 = z/(-1)` do not intersect.

Answer 1

`(x - 1)/1 = (y - 2)/2 = (z + 1)/-1`   ....(1)

and `x/2 = (y - 3)/2 = z/(-1)`     ....(2)

Obviously, the lines are not parallel since their d.r.s. are not proportionate.

Now the coordinates of any variable point on line (1) are (r + 1, 2r + 2, – r – 1) and coordinates of any point on line (2) are (2r’, 2r’ + 3, –r’)

If these points coincide for some r and r' values, then the lines will intersect.

i.e. the lines will intersect if:

r + 1 = 2r’,

2r + 2 = 2r’ + 3,

–r – 1 = – r’

i.e. r – 2r’ = –1  ....(3)

2r – 2r’ = 1   ....(4)

–r + r’ = 1  ....(5)

i.e. the lines will intersect if equations (3), (4) and (5) are consistent.

Now, D = `|(1, -2, -1),(2, -2, 1),(-1, 1, 1)|`

= 1(–2 – 1) + 2(2 + 1) – 1(2 – 2)

= –3 + 6

= 3 ≠ 0

∴ The equations are not consistent,

Hence the lines do not intersect.