The vector \(\vec{a}=α \hat{i}+2\hat{j}+β \hat{k}\) lies in the plane of the vector \(\vec{b}=\hat{i} + \hat{j}\) and \(\vec{c}=\hat{j} + \hat{k}\) and bisects the angle between \(\vec{b}\) and \(\vec{c}\). Then, which one of the following gives possible values of α and β ?

Concept: 

If two or more vectors lie on the same plane than they are called coplanar vector and satisfies the conditions \(\left[ \vec{a}\vec{b}\vec{c} \right]=0\).

\(\left| \begin{matrix} {{a}_{x}} & {{a}_{y}} & {{a}_{z}} \\ {{b}_{x}} & {{b}_{y}} & {{b}_{z}} \\ {{c}_{x}} & {{c}_{y}} & {{c}_{z}} \\ \end{matrix} \right|=0 \)

Calculation: 

\(\left[ \vec{a}\vec{b}\vec{c} \right]=0 \)

\(\left| \begin{matrix} {α} & {2} & {β} \\ {1} & {1} & {0} \\ {0} & {1} & {1} \\ \end{matrix} \right|=0 \)

\(α+β=2\)

Also, \(\vec{a}\) bisects the angle between \(\vec{b}\) and \(\vec{c}\).

\(\begin{align} & \vec{a}=\frac{λ }{√{2}}\left( \hat{b}+\hat{c} \right) \\ & =\frac{λ }{√{2}}\left( \hat{i}+2\hat{j}+\hat{k} \right) \\ \end{align} \)

On comparing with  \(\vec{a}=α \hat{i}+2\hat{j}+β \hat{k}\),

We get, λ = √ 2, α =1, and β = 1