If $l, m, n$ are the direction cosines of a line in three-dimensional space, prove that $l^2 + m^2 + n^2 = 1$. Hence, find the angle between two lines whose direction ratios are $\langle 1, 2, 2 \rangle$ and $\langle 2, -2, 1 \rangle$.

If a line makes angles $\alpha, \beta, \gamma$ with the positive $x, y, z$-axes respectively, prove that: (a) $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ (b) $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$

Find the equation of the plane passing through the three non-collinear points $A(1, 1, 0)$, $B(1, 2, 1)$, and $C(-2, 2, -1)$. Also find the perpendicular distance of this plane from the origin.