Key Concepts and Formulas

• Magnitude Squared of a Vector: For any vector $\vec v$, $|\vec v|^2 = \vec v \cdot \vec v$.
• Expansion of Squared Magnitude of Vector Difference: For any two vectors $\vec x$ and $\vec y$, $|\vec x - \vec y|^2 = |\vec x|^2 + |\vec y|^2 - 2(\vec x \cdot \vec y)$.
• Expansion of Squared Magnitude of Vector Sum: For any two vectors $\vec x$ and $\vec y$ and a scalar $k$, $|\vec x + k\vec y|^2 = |\vec x|^2 + k^2|\vec y|^2 + 2k(\vec x \cdot \vec y)$.
• Unit Vector Property: If $\vec v$ is a unit vector, then $|\vec v| = 1$, which implies $|\vec v|^2 = 1$.

Step-by-Step Solution

Step 1: Expand the given equation using the properties of vector magnitudes. We are given the equation: ${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow a - \overrightarrow c } \right|^2} = 8$. We use the formula $|\vec x - \vec y|^2 = |\vec x|^2 + |\vec y|^2 - 2(\vec x \cdot \vec y)$ for both terms. $(|\overrightarrow a|^2 + |\overrightarrow b|^2 - 2\overrightarrow a \cdot \overrightarrow b) + (|\overrightarrow a|^2 + |\overrightarrow c|^2 - 2\overrightarrow a \cdot \overrightarrow c) = 8$ Reasoning: This step converts the magnitude squared terms into expressions involving dot products and squared magnitudes, which are easier to manipulate.

Step 2: Substitute the magnitudes of the unit vectors. Since $\overrightarrow a$, $\overrightarrow b$, and $\overrightarrow c$ are unit vectors, we have $|\overrightarrow a|^2 = 1$, $|\overrightarrow b|^2 = 1$, and $|\overrightarrow c|^2 = 1$. $(1 + 1 - 2\overrightarrow a \cdot \overrightarrow b) + (1 + 1 - 2\overrightarrow a \cdot \overrightarrow c) = 8$ $2 - 2\overrightarrow a \cdot \overrightarrow b + 2 - 2\overrightarrow a \cdot \overrightarrow c = 8$ Reasoning: Using the unit vector property simplifies the equation by replacing the squared magnitudes with their numerical value of 1.

Step 3: Simplify the equation to find the sum of dot products. Combine the constant terms and rearrange the equation: $4 - 2(\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c) = 8$ $-2(\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c) = 8 - 4$ $-2(\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c) = 4$ $\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c = \frac{4}{-2}$ $\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c = -2$ Reasoning: This step isolates the sum of the dot products $\overrightarrow a \cdot \overrightarrow b$ and $\overrightarrow a \cdot \overrightarrow c$, which will be a key component in evaluating the target expression.

Step 4: Expand the target expression. The target expression is ${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2} + {\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$. We use the formula $|\vec x + k\vec y|^2 = |\vec x|^2 + k^2|\vec y|^2 + 2k(\vec x \cdot \vec y)$ for both terms, with $k=2$. $\left( |\overrightarrow a|^2 + (2)^2|\overrightarrow b|^2 + 2(2)(\overrightarrow a \cdot \overrightarrow b) \right) + \left( |\overrightarrow a|^2 + (2)^2|\overrightarrow c|^2 + 2(2)(\overrightarrow a \cdot \overrightarrow c) \right)$ $= \left( |\overrightarrow a|^2 + 4|\overrightarrow b|^2 + 4\overrightarrow a \cdot \overrightarrow b \right) + \left( |\overrightarrow a|^2 + 4|\overrightarrow c|^2 + 4\overrightarrow a \cdot \overrightarrow c \right)$ Reasoning: This step expands the target expression into terms of squared magnitudes and dot products, similar to Step 1, but for a sum with a scalar multiple.

Step 5: Substitute the magnitudes of the unit vectors into the expanded expression. Again, since $\overrightarrow a$, $\overrightarrow b$, and $\overrightarrow c$ are unit vectors, $|\overrightarrow a|^2 = 1$, $|\overrightarrow b|^2 = 1$, and $|\overrightarrow c|^2 = 1$. $= (1 + 4(1) + 4\overrightarrow a \cdot \overrightarrow b) + (1 + 4(1) + 4\overrightarrow a \cdot \overrightarrow c)$ $= (1 + 4 + 4\overrightarrow a \cdot \overrightarrow b) + (1 + 4 + 4\overrightarrow a \cdot \overrightarrow c)$ $= 5 + 4\overrightarrow a \cdot \overrightarrow b + 5 + 4\overrightarrow a \cdot \overrightarrow c$ Reasoning: Substituting the unit vector properties simplifies the numerical components of the expanded expression.

Step 6: Rearrange and factor the expression. Combine the constant terms and factor out common scalar multiples from the dot product terms: $= (5 + 5) + 4(\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c)$ $= 10 + 4(\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c)$ Reasoning: Grouping terms and factoring makes it easier to substitute the result from Step 3.

Step 7: Substitute the value of $(\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c)$ found in Step 3. From Step 3, we know that $\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c = -2$. $= 10 + 4(-2)$ $= 10 - 8$ $= 2$ Reasoning: This final substitution uses the previously derived relationship to compute the numerical value of the target expression.

Common Mistakes & Tips

• Expansion Errors: Be extremely careful when expanding $|\vec x - \vec y|^2$ and $|\vec x + k\vec y|^2$. Forgetting the $2(\vec x \cdot \vec y)$ term or miscalculating $k^2$ are common errors.
• Unit Vector Property Neglect: Always remember that for unit vectors, $|\vec v|^2 = 1$. This simplifies many parts of the calculation.
• Algebraic Slip-ups: Double-check your arithmetic and algebraic manipulations, especially when combining terms and solving for unknown dot product sums.

Summary

The problem requires expanding vector magnitude expressions and utilizing the properties of unit vectors. We first simplified the given equation involving $|\vec a - \vec b|^2$ and $|\vec a - \vec c|^2$ to find the value of $\overrightarrow a \cdot \overrightarrow b + \overrightarrow a \cdot \overrightarrow c$. Then, we expanded the target expression ${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2} + {\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$, substituted the known unit vector magnitudes, and finally used the derived sum of dot products to arrive at the final numerical answer.

The final answer is \boxed{2}.