Two particles are located at equal distance from origin. The position vectors of those are represented by A⃗=2î+3nĵ+2k̂ and B̄=2î-2ĵ+4pk̂, respectively. If both the vectors are at right angle to each other, the value of n⁻¹ is ________ .

Solution

Physics sir · 4 steps

Two conditions, two unknowns: equal lengths give one equation and the zero dot product gives the other.

1. Equal distances. Equal distance from the origin means |A⃗|² = |B⃗|². So 4 + 9n² + 4 = 4 + 4 + 16p², which gives 9n² = 16p².
2. Right angle. Perpendicular vectors have a zero dot product: A⃗ · B⃗ = 4 - 6n + 8p = 0. So 4p = 3n - 2.

   16p² = (4p)² = (3n-2)² = 9n²

3. Solve for n. Expand to get 9n² - 12n + 4 = 9n². The n² terms cancel, leaving 12n = 4, so n = 1/3 and p = -1/4. Check: both lengths come out as 3, and the dot product is 4 - 2 - 2 = 0.
4. So. n⁻¹ = 3, so the answer is 3.

The common slip: 0.33. Reporting n itself gives 1/3 ≈ 0.33, not 3. The question asks for n⁻¹.

The idea to keep. Equal distance from the origin means equal magnitudes, and a right angle means a zero dot product. Square the dot product equation before you substitute, and the squared terms often cancel.

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