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The vector vec a=""alpha hat i+2 hat ...

The vector ` vec a=""alpha hat i+2 hat j+""beta hat k` lies in the plane of the vectors ` 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 `alpha` and `beta` ? (1) `alpha=""2,beta=""2` (2) `alpha=""1,beta=""2` (3) `alpha=""2,beta=""1` (4) `alpha=""1,beta=""1`

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To solve the problem, we need to find the values of \( \alpha \) and \( \beta \) for the vector \( \vec{a} = \alpha \hat{i} + 2 \hat{j} + \beta \hat{k} \), which lies in the plane of vectors \( \vec{b} = \hat{i} + \hat{j} \) and \( \vec{c} = \hat{j} + \hat{k} \) and bisects the angle between them. ### Step 1: Determine the normal vector to the plane formed by \( \vec{b} \) and \( \vec{c} \) To find the normal vector to the plane formed by \( \vec{b} \) and \( \vec{c} \), we can take the cross product of \( \vec{b} \) and \( \vec{c} \). \[ \vec{b} = \hat{i} + \hat{j} ...
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