Mathematics

Problem

Let

\mathbf{a} = 2\hat{\mathbf{i}} - \hat{\mathbf{j}} + \hat{\mathbf{k}}

\mathbf{b} = \hat{\mathbf{i}} + 2\hat{\mathbf{j}} - \hat{\mathbf{k}}

and

\mathbf{c} = \hat{\mathbf{i}} + \hat{\mathbf{j}} - 2\hat{\mathbf{k}}

be three vectors. A vector in the plane of

\mathbf{b}

and

\mathbf{c}

whose projection on

\mathbf{a}

is of magnitude

\sqrt{\frac{2}{3}}

is a.

2\hat{\mathbf{i}} + 3\hat{\mathbf{j}} - 3\hat{\mathbf{k}}

2\hat{\mathbf{i}} + 3\hat{\mathbf{j}} + 3\hat{\mathbf{k}}

-2\hat{\mathbf{i}} - \hat{\mathbf{j}} + 5\hat{\mathbf{k}}

2\hat{\mathbf{i}} + \hat{\mathbf{j}} + 5\hat{\mathbf{k}}