The vector <span class="math-tex">\(\vec{a}=α \hat{i}+2\hat{j}+β \hat{k}\)</span> lies in the plane of the vector <span class="math-tex">\(\vec{b}=\hat{i} + \hat{j}\)</span> and <span class="math-tex">\(\vec{c}=\hat{j} + \hat{k}\)</span> and bisects the angle between <span class="math-tex">\(\vec{b}\)</span> and <span class="math-tex">\(\vec{c}\)</span>. Then, which one of the following gives possible values of α and β ?
The angle bisector of \(\vec{b}\) and \(\vec{c}\) points along \(\hat{b} + \hat{c}\):
\(\frac{\hat{i} + \hat{j}}{\sqrt{2}} + \frac{\hat{j} + \hat{k}}{\sqrt{2}} \propto \hat{i} + 2\hat{j} + \hat{k}\)
\(\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k}\) must be a multiple of \(\hat{i} + 2\hat{j} + \hat{k}\). The j-component is already 2, so the multiple is 1: \(\alpha = 1\), \(\beta = 1\).
Answer: (D)
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