Q.393. Point of intersection of two lines in 3-D.

Question 393.
Find the point of intersection of the lines represented by
(i + 3j) + λ(- i - j) and (i + 2j) + λ(i - j).

Answer 393.
The given lines are
r (vector) = (1, 3) + λ(-1, -1)
r (vector) = (1, 2) + λ(1, -1)
Their directions are (-1, -1) and (1, -1) and the dot product = -1 + 1 = 0
=> they are perpendicular.
Let (x, y) be their point of intersection
=> x = 1 - λ1 = 1 + λ2
and y = 3 - λ1 = 2 - λ2
=> λ1 + λ2 = 0 and λ1 - λ2 = 1
=> λ1 = 1/2 and λ2 = - 1/2

Plugging λ1 = 1/2 in the eqn. of the first line,
(x, y) = (1/2, 5/2)
Plugging λ2 = -1/2 in the eqn. of the second line,
(x, y) = 1/2, 5/2)

=> (1/2, 5/2) is the common point of the two lines and is the point of intersection.

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