Step 1: Understanding the Concept:
We are given the magnitudes of three vectors and need to find the maximum possible value of a symmetric expression involving their differences. Expanding the squares of the magnitudes will relate the expression to the dot products of the vectors.
Step 2: Key Formula or Approach:
Use the expansion: $|\vec{u} - \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2 - 2(\vec{u} \cdot \vec{v})$.
Use the property that the square of the magnitude of any sum of vectors is non-negative: $|\bar{a} + \bar{b} + \bar{c}|^2 \ge 0$.
Step 3: Detailed Explanation:
Let the given expression be $S$:
\[ S = |\bar{a} - \bar{b}|^2 + |\bar{b} - \bar{c}|^2 + |\bar{c} - \bar{a}|^2 \]
Expand each term:
\[ S = (|\bar{a}|^2 + |\bar{b}|^2 - 2\bar{a}\cdot\bar{b}) + (|\bar{b}|^2 + |\bar{c}|^2 - 2\bar{b}\cdot\bar{c}) + (|\bar{c}|^2 + |\bar{a}|^2 - 2\bar{c}\cdot\bar{a}) \]
Group like terms together:
\[ S = 2(|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2) - 2(\bar{a}\cdot\bar{b} + \bar{b}\cdot\bar{c} + \bar{c}\cdot\bar{a}) \quad \dots \text{ (Equation 1)} \]
To find an inequality, consider the square of the sum of the three vectors:
\[ |\bar{a} + \bar{b} + \bar{c}|^2 \ge 0 \]
Expand this squared term:
\[ |\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2 + 2(\bar{a}\cdot\bar{b} + \bar{b}\cdot\bar{c} + \bar{c}\cdot\bar{a}) \ge 0 \]
Rearrange to isolate the dot product terms:
\[ - 2(\bar{a}\cdot\bar{b} + \bar{b}\cdot\bar{c} + \bar{c}\cdot\bar{a}) \le |\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2 \]
Now, substitute this inequality back into Equation 1:
\[ S \le 2(|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2) + (|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2) \]
\[ S \le 3(|\bar{a}|^2 + |\bar{b}|^2 + |\bar{c}|^2) \]
We are given the magnitudes: $|\bar{a}| = 3$, $|\bar{b}| = 5$, $|\bar{c}| = 7$.
Substitute these values into the inequality:
\[ S \le 3(3^2 + 5^2 + 7^2) \]
\[ S \le 3(9 + 25 + 49) \]
\[ S \le 3(83) \]
\[ S \le 249 \]
Thus, the expression does not exceed 249.
Step 4: Final Answer:
The maximum value is 249.