The unit vector which is orthogonal to the vector and is coplanar with the vectors and is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: C
Solution
A vector coplanar with a = 2î + ĵ + k̂ and b = î - ĵ + k̂ can be written as v = λa + μb = (2λ+μ)î + (λ-μ)ĵ + (λ+μ)k̂.
For orthogonality with n = 3î + 2ĵ + 6k̂:
v·n = 3(2λ+μ) + 2(λ-μ) + 6(λ+μ) = 6λ+3μ + 2λ-2μ + 6λ+6μ = 14λ + 7μ = 0, giving μ = -2λ.
Substituting μ = -2λ: v = (2λ-2λ)î + (λ+2λ)ĵ + (λ-2λ)k̂ = 0î + 3λĵ - λk̂ = λ(3ĵ - k̂).
The magnitude is |λ|√(9+1) = |λ|√10, so the unit vector is (3ĵ - k̂)/√10.
For orthogonality with n = 3î + 2ĵ + 6k̂:
v·n = 3(2λ+μ) + 2(λ-μ) + 6(λ+μ) = 6λ+3μ + 2λ-2μ + 6λ+6μ = 14λ + 7μ = 0, giving μ = -2λ.
Substituting μ = -2λ: v = (2λ-2λ)î + (λ+2λ)ĵ + (λ-2λ)k̂ = 0î + 3λĵ - λk̂ = λ(3ĵ - k̂).
The magnitude is |λ|√(9+1) = |λ|√10, so the unit vector is (3ĵ - k̂)/√10.
[AIEEE 2004]