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No, the cross product is not generally associative. This means that $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ is not necessarily equal to $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$.

Conditions: You are evaluating expressions involving multiple cross products.; You are checking if grouping affects the result.

No, the cross product is not generally associative. This means that $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ is not necessarily equal to $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$.

Conditions: You are evaluating expressions involving multiple cross products.; You are checking if grouping affects the result.

Yes, the cross product distributes over addition. This means $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$ and $(\mathbf{a} + \mathbf{b}) \times \mathbf{c} = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}$.

Conditions: You are simplifying expressions involving cross products and sums.; You are applying algebraic properties to vector operations.

Yes, the cross product distributes over addition. This means $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$ and $(\mathbf{a} + \mathbf{b}) \times \mathbf{c} = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}$.

Conditions: You are simplifying expressions involving cross products and sums.; You are applying algebraic properties to vector operations.