Question

Check whether the lines given by `(x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 4)/5 = (y - 1)/2 = z` are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

Answer 1

Given lines:

`(x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 4)/5 = (y - 1)/2 = z`

For the first line, direction ratios are (2, 3, 4),

For the second line:

`(x - 4)/5 = (y - 1)/2 = (z - 0)/1`, direction ratios are (5, 2, 1),

Since `2/5 ≠ 3/2 ≠ 4/1`, the lines are not parallel.

Now let the first line be:

x = 1 + 2λ, y = 2 + 3λ, z = 3 + 4λ

Second line:

x = 4 + 5μ, y = 1 + 2μ, z = μ

Let’s equate them:

1 + 2λ = 4 + 5μ      ....(1)

2 + 3λ = 1 + 2μ      ....(2)

3 + 4λ = μ      ....(3)

From third equation: μ = 3 + 4λ      ....(4)

Substitute equation (4) in the second:

2 + 3λ = 1 + 2(3 + 4λ)

2 + 3λ = 7 + 8λ

−5 = 5λ

∴ λ = −1

Let’s put λ = −1 in equation (4):

μ = 3 + 4(−1)

μ = 3 + (−4)

∴ μ = −1

Now substitute λ = −1 and μ = −1 in equation (1):

1 + 2(−1) = 4 + 5(−1)

−1 = −1

So lines intersect.

Point of intersection:

• x = 1 + 2(−1) = −1
• y = 2 + 3(−1) = −1
• z = 3 + 4(−1) = −1

Hence, the two lines are not parallel, and they intersect; their point of intersection is (−1, −1, −1).