We are asked to evaluate the following expression: $\frac{-\frac{\sqrt{3}}{3} \times (-2) + \sqrt{\frac{7}{3}} \times (-2) + \frac{\sqrt{3}}{3} \times 1}{\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} \times \sqrt{(-2)^2 + (-2)^2 + (1)^2}}$

1. Problem Description

We are asked to evaluate the following expression:

$\frac{-\frac{\sqrt{3}}{3} \times (-2) + \sqrt{\frac{7}{3}} \times (-2) + \frac{\sqrt{3}}{3} \times 1}{\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} \times \sqrt{(-2)^2 + (-2)^2 + (1)^2}}$

2. Solution Steps

First, simplify the numerator:

$-\frac{\sqrt{3}}{3} \times (-2) = \frac{2\sqrt{3}}{3}$

$\sqrt{\frac{7}{3}} \times (-2) = -2\sqrt{\frac{7}{3}} = -2 \frac{\sqrt{7}}{\sqrt{3}} = -2 \frac{\sqrt{7}\sqrt{3}}{3} = -\frac{2\sqrt{21}}{3}$

$\frac{\sqrt{3}}{3} \times 1 = \frac{\sqrt{3}}{3}$

So, the numerator is:

$\frac{2\sqrt{3}}{3} - \frac{2\sqrt{21}}{3} + \frac{\sqrt{3}}{3} = \frac{3\sqrt{3}}{3} - \frac{2\sqrt{21}}{3} = \sqrt{3} - \frac{2\sqrt{21}}{3}$

Now, simplify the denominator:

$(\frac{\sqrt{3}}{3})^2 = \frac{3}{9} = \frac{1}{3}$

$\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} = \sqrt{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = \sqrt{1} = 1$

$(-2)^2 = 4$

$(1)^2 = 1$

$\sqrt{(-2)^2 + (-2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$

So, the denominator is:

$1 \times 3 = 3$

Therefore, the expression becomes:

$\frac{\sqrt{3} - \frac{2\sqrt{21}}{3}}{3} = \frac{\sqrt{3}}{3} - \frac{2\sqrt{21}}{9} = \frac{3\sqrt{3} - 2\sqrt{21}}{9}$

3. Final Answer

$\frac{3\sqrt{3} - 2\sqrt{21}}{9}$