1. Key Concepts and Formulas

• Vector Triple Product Formula: The identity $(\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{B} \cdot \vec{C})\vec{A}$ is fundamental for simplifying nested cross products.
• Unit Vector Cross Products: The cyclic properties of the standard basis vectors are essential: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$, and their anti-commutative counterparts like $\hat{j} \times \hat{i} = -\hat{k}$.
• Dot Product: The dot product of two vectors $\vec{P} = P_x\hat{i} + P_y\hat{j} + P_z\hat{k}$ and $\vec{Q} = Q_x\hat{i} + Q_y\hat{j} + Q_z\hat{k}$ is given by $\vec{P} \cdot \vec{Q} = P_x Q_x + P_y Q_y + P_z Q_z$.

2. Step-by-Step Solution

We are given the vectors $\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 4\hat{k}$. We need to compute $\vec{c} = (((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$ and then find $\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})$.

Step 1: Simplify the innermost cross product $(\vec{a} \times \vec{b}) \times \hat{i}$ We use the vector triple product formula $(\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{B} \cdot \vec{C})\vec{A}$. Here, $\vec{A} = \vec{a}$, $\vec{B} = \vec{b}$, and $\vec{C} = \hat{i}$. First, calculate the dot products: $\vec{a} \cdot \hat{i} = (-5\hat{i} + \hat{j} - 3\hat{k}) \cdot \hat{i} = -5$. $\vec{b} \cdot \hat{i} = (\hat{i} + 2\hat{j} - 4\hat{k}) \cdot \hat{i} = 1$. Applying the formula: $(\vec{a} \times \vec{b}) \times \hat{i} = (\vec{a} \cdot \hat{i})\vec{b} - (\vec{b} \cdot \hat{i})\vec{a} = (-5)\vec{b} - (1)\vec{a} = -5\vec{b} - \vec{a}$. Substitute the vector components: $-5(\hat{i} + 2\hat{j} - 4\hat{k}) - (-5\hat{i} + \hat{j} - 3\hat{k}) = (-5\hat{i} - 10\hat{j} + 20\hat{k}) - (-5\hat{i} + \hat{j} - 3\hat{k})$ $= -5\hat{i} - 10\hat{j} + 20\hat{k} + 5\hat{i} - \hat{j} + 3\hat{k} = -11\hat{j} + 23\hat{k}$. Let $\vec{X} = -11\hat{j} + 23\hat{k}$. Then $\vec{c} = ((\vec{X} \times \hat{i}) \times \hat{i})$.

Step 2: Simplify the next cross product $\vec{X} \times \hat{i}$ We need to calculate $(-11\hat{j} + 23\hat{k}) \times \hat{i}$. Using the distributive property and unit vector cross products: $\vec{X} \times \hat{i} = -11(\hat{j} \times \hat{i}) + 23(\hat{k} \times \hat{i})$. Since $\hat{j} \times \hat{i} = -\hat{k}$ and $\hat{k} \times \hat{i} = \hat{j}$: $\vec{X} \times \hat{i} = -11(-\hat{k}) + 23(\hat{j}) = 11\hat{k} + 23\hat{j}$. Let $\vec{Y} = 23\hat{j} + 11\hat{k}$. Then $\vec{c} = \vec{Y} \times \hat{i}$.

Step 3: Simplify the outermost cross product $\vec{Y} \times \hat{i}$ We need to calculate $(23\hat{j} + 11\hat{k}) \times \hat{i}$. $\vec{Y} \times \hat{i} = 23(\hat{j} \times \hat{i}) + 11(\hat{k} \times \hat{i})$. Using the unit vector cross product properties again: $\vec{Y} \times \hat{i} = 23(-\hat{k}) + 11(\hat{j}) = -23\hat{k} + 11\hat{j}$. So, $\vec{c} = 11\hat{j} - 23\hat{k}$.

Step 4: Calculate the dot product $\vec{c} \cdot (-\hat{i}+\hat{j}+\hat{k})$ Now we compute the dot product of $\vec{c} = 0\hat{i} + 11\hat{j} - 23\hat{k}$ with $(-\hat{i} + \hat{j} + \hat{k})$. $\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k}) = (0)(-1) + (11)(1) + (-23)(1)$. $= 0 + 11 - 23$. $= -12$.

3. Common Mistakes & Tips

• Order in Vector Triple Product: Be extremely careful with the order of vectors in the formula $(\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{B} \cdot \vec{C})\vec{A}$. Swapping $\vec{A}$ and $\vec{B}$ will lead to an incorrect result.
• Unit Vector Cross Products: Memorize the cyclic order and anti-commutative property of $\hat{i}, \hat{j}, \hat{k}$ to avoid errors in step-by-step cross product calculations.
• Nested Calculations: Work from the innermost cross product outwards to systematically simplify the expression for $\vec{c}$.

4. Summary

The problem required the simplification of a vector $\vec{c}$ defined by a series of nested cross products and then calculating its dot product with another vector. We effectively used the vector triple product formula to break down the complex expression for $\vec{c}$ step-by-step, starting from the innermost cross product. After simplifying $\vec{c}$ to $11\hat{j} - 23\hat{k}$, we performed the dot product calculation, yielding the final result.

5. Final Answer

The value of $\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})$ is $-12$. This corresponds to option (A).

The final answer is $\boxed{\text{-12}}$.