Key Concepts and Formulas

• Vector Triple Product Identity: For any three vectors $\overrightarrow P$, $\overrightarrow Q$, and $\overrightarrow R$, the identity is given by: $\overrightarrow P \times (\overrightarrow Q \times \overrightarrow R) = (\overrightarrow P \cdot \overrightarrow R)\overrightarrow Q - (\overrightarrow P \cdot \overrightarrow Q)\overrightarrow R$
• Dot Product: The dot product of a vector with itself is its magnitude squared: $\overrightarrow v \cdot \overrightarrow v = |\overrightarrow v|^2$.
• Magnitude of a Vector: For a vector $\overrightarrow v = v_x \widehat i + v_y \widehat j + v_z \widehat k$, its magnitude squared is $|\overrightarrow v|^2 = v_x^2 + v_y^2 + v_z^2$.

Step-by-Step Solution

Step 1: Isolate the cross product term We are given the vector equation $\overrightarrow a \times \overrightarrow c + \overrightarrow b = \overrightarrow 0$. To proceed, we isolate the term involving $\overrightarrow c$: $\overrightarrow a \times \overrightarrow c = -\overrightarrow b$ Reasoning: This step rearranges the given equation to isolate the cross product of $\overrightarrow a$ and $\overrightarrow c$, which is a necessary precursor to applying the vector triple product identity.

Step 2: Apply the vector triple product identity To utilize the vector triple product identity and introduce the scalar product $\overrightarrow a \cdot \overrightarrow c$, we take the cross product of both sides of the equation from Step 1 with $\overrightarrow a$: $\overrightarrow a \times (\overrightarrow a \times \overrightarrow c) = \overrightarrow a \times (-\overrightarrow b)$ $\overrightarrow a \times (\overrightarrow a \times \overrightarrow c) = -(\overrightarrow a \times \overrightarrow b)$ Now, we apply the vector triple product identity $\overrightarrow P \times (\overrightarrow Q \times \overrightarrow R) = (\overrightarrow P \cdot \overrightarrow R)\overrightarrow Q - (\overrightarrow P \cdot \overrightarrow Q)\overrightarrow R$ to the left side, with $\overrightarrow P = \overrightarrow a$, $\overrightarrow Q = \overrightarrow a$, and $\overrightarrow R = \overrightarrow c$: $(\overrightarrow a \cdot \overrightarrow c)\overrightarrow a - (\overrightarrow a \cdot \overrightarrow a)\overrightarrow c = -(\overrightarrow a \times \overrightarrow b)$ Reasoning: This is the crucial step where the vector triple product identity transforms a nested cross product into an expression involving dot products and scalar multiples of vectors. This allows us to relate the unknown vector $\overrightarrow c$ to known quantities.

Step 3: Calculate necessary scalar products and cross products We need to evaluate $\overrightarrow a \cdot \overrightarrow a$ and $\overrightarrow a \times \overrightarrow b$. Given $\overrightarrow a = \widehat i - \widehat j$ and $\overrightarrow b = \widehat i + \widehat j + \widehat k$.

1. Calculate $\overrightarrow a \cdot \overrightarrow a$: $\overrightarrow a \cdot \overrightarrow a = |\overrightarrow a|^2 = (1)^2 + (-1)^2 + (0)^2 = 1 + 1 + 0 = 2$
2. Calculate $\overrightarrow a \times \overrightarrow b$: $\overrightarrow a \times \overrightarrow b = \begin{vmatrix} \widehat i & \widehat j & \widehat k \\ 1 & -1 & 0 \\ 1 & 1 & 1 \end{vmatrix} = \widehat i((-1)(1) - (0)(1)) - \widehat j((1)(1) - (0)(1)) + \widehat k((1)(1) - (-1)(1))$ $= \widehat i(-1) - \widehat j(1) + \widehat k(1+1) = -\widehat i - \widehat j + 2\widehat k$ Reasoning: These calculations provide the numerical values needed to substitute into the equation derived in Step 2, simplifying the problem towards solving for $\overrightarrow c$.

Step 4: Substitute known values and solve for $\overrightarrow c$ Substitute the calculated values and the given $\overrightarrow a \cdot \overrightarrow c = 4$ into the equation from Step 2: $(4)\overrightarrow a - (2)\overrightarrow c = -(-\widehat i - \widehat j + 2\widehat k)$ $4\overrightarrow a - 2\overrightarrow c = \widehat i + \widehat j - 2\widehat k$ Substitute $\overrightarrow a = \widehat i - \widehat j$: $4(\widehat i - \widehat j) - 2\overrightarrow c = \widehat i + \widehat j - 2\widehat k$ $4\widehat i - 4\widehat j - 2\overrightarrow c = \widehat i + \widehat j - 2\widehat k$ Rearrange to solve for $2\overrightarrow c$: $2\overrightarrow c = (4\widehat i - 4\widehat j) - (\widehat i + \widehat j - 2\widehat k)$ $2\overrightarrow c = (4-1)\widehat i + (-4-1)\widehat j - (-2)\widehat k$ $2\overrightarrow c = 3\widehat i - 5\widehat j + 2\widehat k$ Now, solve for $\overrightarrow c$: $\overrightarrow c = \frac{1}{2}(3\widehat i - 5\widehat j + 2\widehat k) = \frac{3}{2}\widehat i - \frac{5}{2}\widehat j + \widehat k$ Reasoning: By substituting all known scalar and vector quantities, we obtain a linear equation for $\overrightarrow c$, which can be solved by algebraic manipulation to find its component form.

Step 5: Calculate $|\overrightarrow c|^2$ We need to find the magnitude squared of $\overrightarrow c$. Using the components of $\overrightarrow c = \frac{3}{2}\widehat i - \frac{5}{2}\widehat j + \widehat k$: $|\overrightarrow c|^2 = \left(\frac{3}{2}\right)^2 + \left(-\frac{5}{2}\right)^2 + (1)^2$ $|\overrightarrow c|^2 = \frac{9}{4} + \frac{25}{4} + 1$ $|\overrightarrow c|^2 = \frac{9+25}{4} + 1$ $|\overrightarrow c|^2 = \frac{34}{4} + 1$ $|\overrightarrow c|^2 = \frac{17}{2} + \frac{2}{2}$ $|\overrightarrow c|^2 = \frac{19}{2}$ Reasoning: This is the final calculation to answer the question. The magnitude squared of a vector is the sum of the squares of its components.

Common Mistakes & Tips

• Vector Triple Product Order: Ensure the correct order of vectors in the identity $\overrightarrow P \times (\overrightarrow Q \times \overrightarrow R) = (\overrightarrow P \cdot \overrightarrow R)\overrightarrow Q - (\overrightarrow P \cdot \overrightarrow Q)\overrightarrow R$. Incorrectly assigning dot products can lead to wrong signs or vector directions.
• Algebraic Precision: Pay close attention to signs and arithmetic when manipulating vector equations and combining components. A small error can propagate through the solution.
• Cross Product Calculation: Double-check the calculation of the cross product $\overrightarrow a \times \overrightarrow b$, as this value is used in the main equation.

Summary

The problem was solved by first isolating the cross product term $\overrightarrow a \times \overrightarrow c$ from the given vector equation. Then, the vector triple product identity was applied by taking the cross product of both sides with $\overrightarrow a$. This transformed the equation into one involving dot products, which were then calculated. Substituting these values and the given $\overrightarrow a \cdot \overrightarrow c = 4$ allowed us to solve for the unknown vector $\overrightarrow c$. Finally, the magnitude squared of $\overrightarrow c$ was computed using its component form.

The final answer is $\boxed{\frac{19}{2}}$.