Question

Consider the pair of lines \(\bar r\) = 3i + 4j – 2k + λ(-i + 2j + k) \(\bar L_1\)_{, \(\bar r\)} = i – 7j – 2k + µ(i + 3j + 2k) \(\bar L_2\)

1. Find one point each on lines L₁ and L₂. 
2. Find the distance between those points. 
3. Find the shortest distance between L₁ and L₂.

Answer 1

1. By putting λ = 0 in line L₁ and µ = 0 in L₂ we get the required points. L₁ ⇒ \(\bar r\) = 3i + 4j – 2k

∴ Co-ordinate is (3, 4, -2)
L₂ ⇒ \(\bar r\) = i – 7j – 2k
∴ Co-ordinate is (1, -7, -2).

2. Distance between (3, 4, -2) and (1, -7, -2)

3. Let L₁ ⇒ \(\bar r\) = 3i + 4j – 2k + λ(-i + 2j + k) is of the form \(\bar r\) = \(\bar a_1\) +λ\(\bar b_1\) where