Three lines L 1 : r → = λ i ^ ,   λ ∈ R , L 2 : r → = k ^ + μ j ^ ,   μ ∈ R and L 3 :   r → = i ^ + j ^ + ν k ^ ,   ν ∈ R are given. For which point(s) Q and L 2 can we find a point P on L 1 and a point R on L 3 so that P ,   Q and R are collinear?
D. k ^ - 1 2 j ^
As given L 1 ⇒ r → = λ i ,   λ ∈ R L 2 ⇒ r → = k + μ j ,   μ ∈ R and L 3 ⇒ r → = i + j + ν k ,   ν ∈ R Let point P λ ,   0 ,   0 ,   Q 0 ,   μ ,   1 and R 1 ,   1 ,   ν and P Q → = - λ i ^ + μ j ^ + k ^ ,   P R → = 1 - λ i ^ + j ^ + ν k ^ as P ,   Q and R are co-linear ⇒ P Q → is parallel to P R → ⇒ - λ 1 - λ = μ 1 = 1 ν ⇒ λ = μ μ - 1 and ν = 1 μ hence μ ≠ 0 μ ≠ 1 Q ≠ 0 ,   0 ,   1 and Q ≠ 0 ,   1 ,   1 Q ≠ k ^   &   Q ≠ j ^ + k ^
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks