The expression you've written seems a bit unclear due to the formatting. However, based on common vector operations and the given context of non-zero, non-planar vectors a, b, and c, I'm interpreting the expression as:

$\left( \left( \mathbf{a} \mathbf{b} \mathbf{c} \right) \times \left( \mathbf{a} \mathbf{b} \mathbf{a} \right) \right) \cdot \left( \mathbf{c} \right)$

Or something along those lines, involving dot products and cross products.

But before I proceed, it would help if you could clarify the exact format of the expression. If you're referring to a specific determinant, triple product, or another vector identity, please confirm.

I will now assume that you are asking about the scalar triple product and its properties:

Scalar Triple Product:

Given three vectors a, b, and c, the scalar triple product is written as:

$(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}$

This represents the volume of the parallelepiped formed by the vectors a, b, and c. Importantly, this product is zero if the vectors are coplanar and nonzero if the vectors are non-coplanar (i.e., they form a 3D volume).

Formula Evaluation:

If we consider your expression to be a combination of scalar triple product terms, the value will depend on the specific context of the vector operations.

For non-zero, non-coplanar vectors, the answer could potentially involve:

$\mathbf{a} \times \mathbf{b} \times \mathbf{c} \quad \text{and related cross and dot products.}$

But to offer an exact result, clarification is needed on the symbols used. Would you like me to interpret it as a specific formula (determinant, scalar triple product, etc.)?

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Would you like me to clarify more details about vector identities or properties like the scalar triple product?

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Here are 5 related questions for further exploration:

1. What is the geometric meaning of the scalar triple product?
2. How do the cross product and dot product behave geometrically in 3D space?
3. What is the volume of a parallelepiped formed by three vectors?
4. Can you explain the properties of non-coplanar vectors in 3D geometry?
5. How does the distributive property of the cross product work in vector algebra?

Tip: Always check for vector dimensions and their geometric interpretations to understand the underlying relationships between the operations!