If the shortest distance between the lines r₁⃗ = α î + 2ĵ + 2k̂ + λ (î - 2ĵ + 2k̂), λ ∈ R, α > 0 and r₂⃗ = - 4î - k̂ + μ (3î - 2ĵ - 2k̂), μ ∈ R is 9, then α is equal to ____________.

Solution

Maths sir · 4 steps

Take the cross product of the two directions, project the joining vector onto it, and set that equal to 9.

1. Read off the lines. Line 1 passes through A(α, 2, 2) with direction b⃗₁ = (1, -2, 2). Line 2 passes through B(-4, 0, -1) with direction b⃗₂ = (3, -2, -2).
2. Common perpendicular. b⃗₁ × b⃗₂ = (4+4, -(-2-6), -2+6) = (8, 8, 4), which is parallel to n⃗ = (2, 2, 1) with |n⃗| = 3. The joining vector is AB⃗ or BA⃗; take A - B = (α + 4, 2, 3).

   d = (|(A-B)· n⃗|)/|n⃗| = (|2(α+4) + 4 + 3|)/3 = (|2α + 15|)/3

3. Set it to 9. |2α + 15| = 27, so 2α + 15 = 27 or 2α + 15 = -27. That gives α = 6 or α = -21.
4. So. The question says α > 0, so α = 6.

The common slip: -21. Keeping the negative root of |2α + 15| = 27 gives -21, which the condition α > 0 rules out.

The idea to keep. Shortest distance between skew lines is the projection of any joining vector onto b⃗₁ × b⃗₂. Scale the cross product down to small numbers first, and remember the modulus gives two roots.

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