A vector a→$$=$$α$$i^+2j^+$$β$$k^$$, 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→ andc→, . then

Question

A vector  a→$$=$$α$$i^+2j^+$$β$$k^$$, 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→ andc→, . then

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Angle bisector can be  a→$$=$$λ$$b^+c^$$  or   a→$$=$$μ$$b^$$−$$c^Now$$ a→$$=$$λ$$i^+j^2+i^-j^+4k^32=$$λ$$323i^+3j^+i^$$−$$j^+4k^=$$λ$$324i^+2j^+4k^$$          Comparing  with a→$$=$$α$$i^+2j^+$$β$$k^$$ we get                                               $$2$$λ$$32=2$$⇒λ$$=32$$          ⇒a→$$=4i^+2j^+4k^$$                   None of the option is satisfied. So, now considera→$$=$$μ$$i^+j^2$$−$$i^$$−$$j^+4k^32$$      ⇒a→$$=$$μ$$323i^+3j^$$−$$i^+j^$$−$$4k^$$               $$=$$μ$$322i^+4j^$$−$$4k^$$                               Comparing witha→$$=$$α$$i^+2j^+$$β$$k^$$, we get    $$4$$μ$$32=2$$⇒μ$$=322$$⇒a→$$=i^+2j^$$−$$2k^Now$$ a→.$$k^+2=-2+2=0$$

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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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→ andc→, . then

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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