Let a line l pass through the origin and be perpendicular to the lines
l₁: r⃗=(î-11 ĵ-7 k̂)+λ(î+2 ĵ+3 k̂), λ ∈ ℝ and
l₂: r⃗=(-î+k̂)+μ(2 î+2 ĵ+k̂), μ ∈ ℝ.
If P is the point of intersection of l and l₁, and Q(∝, β, γ) is the foot of perpendicular from P on l₂, then 9(α+β+γ) is equal to _____________.

Solution

Maths sir · 4 steps

Get the direction of l from a cross product, find P on l₁, then drop the perpendicular onto l₂.

1. Direction of l. A line perpendicular to both l₁ and l₂ runs along the cross product of their directions. (1, 2, 3) × (2, 2, 1) = (2 - 6, 6 - 1, 2 - 4) = (-4, 5, -2). Since l passes through the origin, its points are t(-4, 5, -2).
2. Find P. A point of l₁ is (1 + λ, -11 + 2λ, -7 + 3λ). Equating it to (-4t, 5t, -2t), the first two coordinates give t = -1 and λ = 3, and the third checks out: -7 + 9 = 2. So P = (4, -5, 2).
3. Foot on l2. Take Q = (-1 + 2μ, 2μ, 1 + μ) on l₂. Then PQ⃗ = (2μ - 5, 2μ + 5, μ - 1) must be perpendicular to (2, 2, 1).

   2(2μ - 5) + 2(2μ + 5) + (μ - 1) = 9μ - 1 = 0 ⇒ μ = 1/9

4. So. Q = (-7/9, 2/9, 10/9), so α + β + γ = 5/9 and 9(α + β + γ) = 5.

The idea to keep. Perpendicular to two lines means along the cross product of their directions. For a foot of perpendicular, write the general point on the line and set its dot product with the line's direction to zero.

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