A vector a → = α i ^ + 2 j ^ + β k ^ α , β ∈ R lies in the plane of the…

A vector a→=αi^+2j^+βk^α,β∈R lies in the plane of the vectors, b→=i^+j^ and c→=i^-j^+4k^. If a→ bisects the angle between b→ and c→, then
  1. a→⋅i^+3=0
  2. a→⋅i^+1=0
  3. a→⋅k^+2=0
  4. a→⋅k^+4=0

Solution

As, a→ lies in the plane of the vectors b→ and c→, So, a→=λi^+j^2+i^+j^+4k^32=λ323i^+3j^+i^-j^+4k^=λ324i^+2j^+4k^
Compare with a→=ai^+2j^+βk^
2λ32=2⇒λ=32
a→=4i^+2j^+4k^
Not in option
So, now consider a→=μi^+j^2-i^-j^+4k^32
a→=μ323i^+3j^-i^+j^-4k^
=μ322i^+4j^-4k^
Compare with a→=αi^+2j^+βk^
4μ32=2⇒μ=322
a→=i^+2j^-2k^
a→.k^+2=0

Asked in: JEE Main 2020 (07 Jan Shift 1)

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