Question

Show that the lines vector r = (i + j - k) + λ(3i - j) and vector r = (4i - k) + μ(2i + 3k)  intersect. Find the point of intersection.

Answer 1

The position vectors of arbitrary points on the given lines are

(i + j - k) + λ(3i - j) = (3λ + 1)i + (1 - λ)j - k

and

If the lines intersect, then they have a common point. So, for some values of λ and μ, we must have

Solving the last two of these three equations, we get λ = 1 and μ = 0. These values of λ and m satisfy the first equation. So, the given lines intersect. Putting λ = 1 in first line, we get

vector r = (i + j - k) + (3i - j) = 4i + 0j - k

which is the position vector of the point of intersection. Thus, the coordinates of the point of intersection are (4, 0, −1).