Let x⃗ be a vector in the plane containing vectors a⃗ = 2î - ĵ + k̂ and b⃗ = î + 2ĵ - k̂. If the vector x⃗ is perpendicular to ( 3î + 2ĵ - k̂ ) and its projection on a⃗ is 17√6/2, then the value of |x⃗ |² is equal to __________.

Solution

Maths sir · 5 steps

Write x as a mix of a and b, then let the two conditions fix the two unknowns.

1. Set up x. Any vector in the plane of a⃗ and b⃗ is x⃗ = λ a⃗ + μ b⃗ = (2λ + μ)î + (-λ + 2μ)ĵ + (λ - μ)k̂. Now you need two equations for λ and μ.
2. Perpendicular condition. Take the dot product with 3î + 2ĵ - k̂ and set it to zero: 3(2λ + μ) + 2(-λ + 2μ) - (λ - μ) = 0. This simplifies to 3λ + 8μ = 0.
3. Projection condition. The projection on a⃗ is (x⃗ · a⃗)/|a⃗|, and |a⃗| = √6. So x⃗ · a⃗ = 17√6/2 × √6 = 51. Since a⃗ · a⃗ = 6 and a⃗ · b⃗ = 2 - 2 - 1 = -1, this gives 6λ - μ = 51.

   3λ + 8μ = 0, 6λ - μ = 51 ⇒ λ = 8, μ = -3

4. Find x. x⃗ = 8a⃗ - 3b⃗ = 13î - 14ĵ + 11k̂. Check: 39 - 28 - 11 = 0, so it is perpendicular, and 26 + 14 + 11 = 51, so the projection is right.
5. So. |x⃗|² = 13² + 14² + 11² = 169 + 196 + 121 = 486.

The idea to keep. A vector in the plane of a⃗ and b⃗ is always λ a⃗ + μ b⃗. Two unknowns need two conditions, and each dot-product condition gives you one linear equation.

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