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

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Correct answer: (C) a→⋅k^+2=0. As, a→ lies in the plane of the vectors b→ and c→, So, a→=λ(i^+j^)/(√2)+(i^+j^+4k^)/(3√2)=(λ)/(3√2)3i^+3j^+i^-j^+4k^=(λ)/(3√2)4i^+2j^+4k^ Compare with a→=ai^+2j^+βk^ (2λ)/(3√2)=2⇒λ=3√2 a→=4i^+2j^+4k^ Not in option So, now consider a→=μ(i^+j^√2-i^-j^+4k^)/(3√2) a→=(μ)/(3√2)3i^+3j^-i^+j^-4k^ =(μ)/(3√2)2i^+4j^-4k^ Compare with a→=αi^+2j^+βk^ (4μ)/(3√2)=2⇒μ=(3√2)/2 a→=i^+2j^-2k^ a→.k^+2=0

A vector $\vec{a} = α \hat{i} + 2 \hat{j} + β \hat{k} (α, β \in R)$ lies in the plane of the vectors, $\vec{b} = \hat{i} + \hat{j}$ and $\vec{c} = \hat{i} - \hat{j} + 4 \hat{k} .$ If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c},$ then

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