A line l passing through origin is perpendicular to the lines l 1 : r → = 3 + t i ^ + - 1 + 2 t j ^ + 4 + 2 t k ^ l 2 : r → = 3 + 2 s i ^ + 3 + 2 s j ^ + 2 + s k ^ If the co-ordinates of the point in the first octant on l 2 at a distance of 17 from the point of intersection of l and l 1 are a , b , c , then 18 a + b + c is equal to ___ .

Step-by-step solution

l 1 : r → = 3 + t i ^ + - 1 + 2 t j ^ + 4 + 2 t k ^ l 2 : r → = 3 + 2 s i ^ + 3 + 2 s j ^ + 2 + s k ^ DR of l 1 ≡ 1 , 2 , 2 DR of l 2 ≡ 2 , 2 , 1 l 1 × l 2 = i j k 1 2 2 2 2 1 = i 2 - 4 - j 1 - 4 + k 2 - 4 = - 2 i + 3 j - 2 k Hence, DR of l ( line ⊥ to l 1   &   l 2 ) = - 2 , 3 , - 2 ∴   l : r → = - 2 μ i ^ + 3 μ j ^ - 2 μ k ^ for intersection of l   &   l 1 3 + t = - 2 μ - 1 + 2 t = 3 μ 4 + 2 t = - 2 μ &#8