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Let \( \quad \mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\sqrt{2} \hat{\mathbf{k}}, \mathbf{b}=b_{1} \hat{\mathbf{i}}+b_{2} \hat{\mathbf{j}}+\sqrt{2} \hat{\mathbf{k}} \quad \) and \( \mathbf{c}=5 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\sqrt{2} \hat{\mathbf{k}} \) be three vectors such that the
\( \mathrm{P} \) projection vector of \( \mathbf{b} \) on \( \mathbf{a} \) is \( |\mathbf{a}| \) If \( \mathbf{a}+\mathbf{b} \) is

W perpendicular to \( \mathbf{c} \), then \( |\mathbf{b}| \) is equal to
(2019 Main, 9 Jan II)
(a) 6
(b) 4
(c) \( \sqrt{22} \)
(d) \( \sqrt{32} \)
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