Let L₁: r⃗=(î-ĵ+2 k̂)+λ(î-ĵ+2 k̂), λ ∈ ℝ,
L₂: r⃗=(ĵ-k̂)+μ(3 î+ĵ+p k̂), μ ∈ ℝ , and L₃: r⃗=δ(ℓ î+m ĵ+n k̂), δ ∈ ℝ
be three lines such that L₁ is perpendicular to L₂ and L₃ is perpendicular to both L₁ and L₂. Then, the point which lies on L₃ is

Solution

Maths sir · 4 steps

Fix p from the dot product, then cross the two directions to get the direction of L3.

1. Find p. The directions are (1,-1,2) for L₁ and (3,1,p) for L₂. Perpendicular means their dot product is zero: 3 - 1 + 2p = 0, so p = -1.
2. Direction of L3. A line perpendicular to both L₁ and L₂ runs along the cross product of their directions.

   |î, ĵ, k̂; 1, -1, 2; 3, 1, -1| = (1-2)î - (-1-6)ĵ + (1+3)k̂ = -î + 7ĵ + 4k̂

3. Points on L3. L₃ has no fixed-point term, so it passes through the origin. Its points are δ(-1, 7, 4) for any real δ.
4. So. Taking δ = 1 gives (-1, 7, 4), which is option (c).

The idea to keep. Perpendicular to two lines means along the cross product of their directions. A line with no fixed-point term passes through the origin, so its points are multiples of its direction.

Answer from the NTA final answer key. Solution by PrepWiser.