Question

Prove that the line through points А(0, −1, −1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(−4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.

Answer 1

Line through A(0, −1, −1), B(4, 5, 1):

`(x - 0)/4 = (y + 1)/6 = (z + 1)/2 = lambda`

So,

x = 4λ, y = −1 + 6λ, z = −1 + 2λ

Line through C(3, 9, 4), D(−4, 4, 4):

`(x - 3)/-7 = (y - 9)/-5 = (z - 4)/0 = mu`

So, x = 3 − 7μ, y = 9 − 5μ, z = 4

At the intersection:

−1 + 2λ = 4

2λ = 5

`lambda = 5/2`

Then from line AB:

`x = 4 * 5/2 = 10`

`y = -1 + 6 * 5/2 = 14`

z = 4

So point of intersection is: P(10, 14, 4)

Check on the line CD:

10 = 3 − 7μ

7 = −7μ

μ = −1

y = 9 − 5(−1)

= 14,

z = 4

Hence, the lines intersect at: (10, 14, 4)

Line through origin O(0, 0, 0) and P(10, 14, 4):

Direction ratios are: 10, 14, 4

Dividing by 2: 5, 7, 2

Therefore, the required line is:

`x/5 = y/7 = z/2`