Key Concepts and Formulas

• Vector Triple Product Expansion: The vector triple product can be expanded using the following identities:
  • $(\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{B} \cdot \vec{C})\vec{A}$
  • $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}$
• Collinearity of Vectors: Two non-zero vectors $\vec{u}$ and $\vec{v}$ are parallel if and only if $\vec{u} = k\vec{v}$ for some non-zero scalar $k$.

Step-by-Step Solution

Step 1: Apply Vector Triple Product Expansion to both sides of the given equation. We are given the equation $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$. Using the vector triple product expansion formulas: The left side is $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$. The right side is $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$. Reasoning: This step directly applies the fundamental identities for vector triple products to simplify both sides of the given vector equation.

Step 2: Equate the expanded forms and simplify the resulting equation. Setting the expanded forms equal to each other: $(\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ Subtract $(\vec{a} \cdot \vec{c})\vec{b}$ from both sides: $- (\vec{b} \cdot \vec{c})\vec{a} = - (\vec{a} \cdot \vec{b})\vec{c}$ Multiply both sides by $-1$: $(\vec{b} \cdot \vec{c})\vec{a} = (\vec{a} \cdot \vec{b})\vec{c}$ Reasoning: This algebraic manipulation simplifies the equation by cancelling common terms and rearranging the remaining terms to isolate the relationship between $\vec{a}$ and $\vec{c}$.

Step 3: Analyze the simplified equation using the given conditions. The simplified equation is $(\vec{b} \cdot \vec{c})\vec{a} = (\vec{a} \cdot \vec{b})\vec{c}$. We are given that $\vec{a} \cdot \vec{b} \neq 0$ and $\vec{b} \cdot \vec{c} \neq 0$. Let $k_1 = \vec{b} \cdot \vec{c}$ and $k_2 = \vec{a} \cdot \vec{b}$. From the given conditions, $k_1 \neq 0$ and $k_2 \neq 0$. The equation becomes $k_1 \vec{a} = k_2 \vec{c}$. This can be rewritten as $\vec{a} = \frac{k_2}{k_1} \vec{c}$. Since $k_1$ and $k_2$ are non-zero scalars, the ratio $\frac{k_2}{k_1}$ is also a non-zero scalar. Also, if $\vec{a}$ or $\vec{c}$ were the zero vector, then $\vec{a} \cdot \vec{b}$ or $\vec{b} \cdot \vec{c}$ would be zero, which contradicts the given conditions. Thus, $\vec{a}$ and $\vec{c}$ are non-zero vectors. Reasoning: By introducing scalar variables for the dot products and using the given non-zero conditions, we establish that $\vec{a}$ is a non-zero scalar multiple of $\vec{c}$. This is the definition of parallel vectors.

Step 4: Conclude the relationship between $\vec{a}$ and $\vec{c}$. Since $\vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{c}}\right) \vec{c}$, and $\frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{c}}$ is a non-zero scalar, vectors $\vec{a}$ and $\vec{c}$ are parallel. Reasoning: The derived relationship directly implies that the vectors $\vec{a}$ and $\vec{c}$ are in the same or opposite direction, meaning they are parallel.

Common Mistakes & Tips

• Incorrect Vector Triple Product Formula: Ensure the correct form of the vector triple product is used. The order of vectors in the dot products is crucial.
• Ignoring Conditions: The conditions $\vec{a} \cdot \vec{b} \neq 0$ and $\vec{b} \cdot \vec{c} \neq 0$ are vital. If they were zero, the conclusion about parallelism might not hold. For instance, if $\vec{a} \cdot \vec{b} = 0$, the equation would simplify differently.
• Confusing Parallel and Perpendicular: Parallel vectors have an angle of $0$ or $\pi$ between them, while perpendicular vectors have an angle of $\pi/2$. The derived relationship $\vec{a} = k\vec{c}$ indicates parallelism.

Summary

The problem requires the application of the vector triple product identities. By expanding both sides of the given equation $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$, we arrive at $(\vec{b} \cdot \vec{c})\vec{a} = (\vec{a} \cdot \vec{b})\vec{c}$. Given that $\vec{a} \cdot \vec{b} \neq 0$ and $\vec{b} \cdot \vec{c} \neq 0$, this equation implies that $\vec{a}$ is a non-zero scalar multiple of $\vec{c}$, meaning $\vec{a}$ and $\vec{c}$ are parallel.

The final answer is $\boxed{parallel}$ which corresponds to option (D).