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

b^+c^=i+j2+i+j+4k32=132(4i+4j+4k) =232(2i+2j+2k) α=2 β=2 no option satisfied b^-c^=i+j2-i+j+4k32=132(2i+2j-4k) α=2 β=-4

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

b^+c^=i+j2+i+j+4k32=132(4i+4j+4k) =232(2i+2j+2k) α=2 β=2 no option satisfied b^-c^=i+j2-i+j+4k32=132(2i+2j-4k) α=2 β=-4

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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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$\vec{a} . \hat{i} + 3 = 0$

$\vec{a} . \hat{i} + 1 = 0$

$\vec{a} . \hat{i} + 4 = 0$

$\vec{a} . \hat{k} + 2 = 0$

answer is D.

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Detailed Solution

$\hat{b} + \hat{c} = \frac{i + j}{\sqrt{2}} + \frac{i + j + 4 k}{3 \sqrt{2}} = \frac{1}{3 \sqrt{2}} (4 i + 4 j + 4 k) = \frac{2}{3 \sqrt{2}} (2 i + 2 j + 2 k) α = 2 β = 2 no option satisfied \hat{b} - \hat{c} = \frac{i + j}{2} - (\frac{i + j + 4 k}{3 \sqrt{2}}) = \frac{1}{3 \sqrt{2}} (2 i + 2 j - 4 k) α = 2 β = - 4$