Linear Algebra: Orthogonality and Projections

Let B be a set of vectors $v_1, v_2, \ldots, v_k$ such that all vectors in B have length 1 $(\|\|v_i\|\| = 1)$ for all $i$ and are orthogonal to each other $(v_i \cdot v_j = 0)$ for $i \ne j$. Show that B is an orthonormal set and prove that B is also linearly independent.

Given the vectors $v_1 = \left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$ and $v_2 = \left(\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}\right)$, determine if they form an orthonormal set in $\mathbb{R}^3$.