Question: A unit cube lies in the first octant (non-negative coordinates), with a vertex at the origin \(O\). Let \(A=(1,1,1), B=(1,0,0)\) and \(M\) be the midpoint of the line segment \(\overline{A B}\).

1. Express the vectors \(\overrightarrow{A B}, \overrightarrow{O M}, \overrightarrow{O A}\), and \(\overrightarrow{A O}\) in terms of \(\mathbf{i}, \mathbf{j}\), and \(\mathbf{k}\).
2. Find the cosine of the angle between \(\overrightarrow{O M}\), and \(\overrightarrow{O A}\).
3. Find an equation for the plane containing $O, A$, and $M .$ Is this different from the plane containing $O, A$, and $B ?$

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