The angle θ between the vectors \(\rm \vec{a} = 5\hat{i} - \hat{j} + \hat{k}\) and \(\rm \vec{b} = \hat{i} + \hat{j} - \hat{k}\) equals to
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: