Types of Products Among Three Vectors

Among three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, the different types of vector products are:

1. Dot Product ($\cdot$): Scalar result (e.g., $\mathbf{a} \cdot \mathbf{b}$).
2. Cross Product ($\times$): Vector result (e.g., $\mathbf{a} \times \mathbf{b}$).
3. Scalar Triple Product ($\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$): Scalar result.
4. Vector Triple Product ($(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$): Vector result.

Hence, there are four distinct types of products that can be formed among three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$.

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Proof of $\mathbf{a} * (\mathbf{b}^{-1} \cdot \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$

Here’s the interpretation:

• $\mathbf{b}^{-1}$ is the reciprocal lattice vector corresponding to $\mathbf{b}$. In reciprocal lattice notation, $\mathbf{b}^{-1} \cdot \mathbf{c}$ gives a scalar value representing a reciprocal scalar operation.
• The operator $*$ is defined as multiplication of $\mathbf{a}$ with this scalar.

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Proof Steps:

1. Assume $\mathbf{b}^{-1}$ satisfies the reciprocal condition: $\mathbf{b}^{-1} \cdot \mathbf{b} = 1$ and is orthogonal to other vectors in the lattice, i.e., $\mathbf{b}^{-1} \cdot \mathbf{c} = scalar \; S.$

2. Substituting into the operation: $\mathbf{a} * (\mathbf{b}^{-1} \cdot \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.$

   This works because the scalar $(\mathbf{b}^{-1} \cdot \mathbf{c})$ scales the individual contributions of $\mathbf{a}$ interacting with $\mathbf{b}$ and $\mathbf{c}$.

3. Expand $\mathbf{a} * (\mathbf{b}^{-1} \cdot \mathbf{c})$: $(\mathbf{b}^{-1} \cdot \mathbf{c}) scales = S(\mathbf{a}) = (\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c})$

Thus, the operation holds as per reciprocal lattice definitions.

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Would you like further explanations or step-by-step vector properties proof?

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5 Related Questions:

1. What are the differences between scalar triple and vector triple products?
2. Can you explain the physical interpretation of reciprocal lattice vectors in crystallography?
3. How does the property $\mathbf{b}^{-1} \cdot \mathbf{b} = 1$ apply in real-world vector operations?
4. What are other examples where reciprocal systems are used in physics and mathematics?
5. How can we generalize the operation $*$ for n-dimensional vector systems?

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Tip:

When working with reciprocal lattice vectors, always verify the orthogonality and scaling conditions to ensure valid results.