Let α⃗ = 4î + 3ĵ + 5k̂ and β⃗ = î + 2ĵ - 4k̂. Let β⃗ ₁ be parallel to α⃗ and β⃗ ₂ be perpendicular to α⃗. If β⃗ = β⃗ ₁ + β⃗ ₂, then the value of 5β⃗ ₂ .( î + ĵ + k̂ ) is :

Solution

Maths sir · 4 steps

Project β⃗ onto α⃗ for the parallel part; whatever is left is perpendicular.

1. Parallel part. The part of β⃗ along α⃗ is its projection. Then β⃗₂ = β⃗ - β⃗₁ is automatically perpendicular to α⃗.

   β⃗₁ = (α⃗·β⃗)/|α⃗|² α⃗

2. Put in numbers. α⃗·β⃗ = 4 + 6 - 20 = -10 and |α⃗|² = 16 + 9 + 25 = 50. So β⃗₁ = -1/5α⃗ = -4/5î - 3/5ĵ - k̂.
3. Perpendicular part. β⃗₂ = β⃗ - β⃗₁ = 9/5î + 13/5ĵ - 3k̂, so 5β⃗₂ = 9î + 13ĵ - 15k̂. As a check, α⃗· 5β⃗₂ = 36 + 39 - 75 = 0.
4. So. 5β⃗₂·(î + ĵ + k̂) = 9 + 13 - 15 = 7, which is option (b).

The idea to keep. To split a vector into parts along and across another vector, find the projection first. The leftover is perpendicular, and a quick dot product confirms it.

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