Components and addition of vectors in 2D and 3DMedium

Vector Components along and Perpendicular

If the components of $\vec a = \alpha \hat i + \beta \hat j + \gamma \hat k$ along and perpendicular to $\vec b = 3\hat i + \hat j-\hat k$ respectively, are $\frac{{16}}{{11}}\left( {3\hat i + \hat j-\hat k} \right)$ and $\frac{1}{{11}}\left( {-4\hat i-5\hat j-17\hat k} \right)$ , then α² + β² + γ² is equal to :

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