1. Let's prove the first identity: $$(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c})$$ This is a standard vector identity involving dot and cross products. 2. The formula used is the scalar quadruple product identity: $$ (\mathbf{u} \times \mathbf{v}) \cdot (\mathbf{w} \times \mathbf{x}) = (\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{x}) - (\mathbf{u} \cdot \mathbf{x})(\mathbf{v} \cdot \mathbf{w}) $$ 3. Applying this to vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$, we get the desired identity. 4. For the second identity: $$(\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) = [\mathbf{a}, \mathbf{b}, \mathbf{d}] \mathbf{c} - [\mathbf{a}, \mathbf{b}, \mathbf{c}] \mathbf{d}$$ where $[\mathbf{a}, \mathbf{b}, \mathbf{c}]$ is the scalar triple product $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$. 5. The vector triple product identity is: $$ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} $$ 6. Using this, set $\mathbf{u} = \mathbf{a} \times \mathbf{b}$, $\mathbf{v} = \mathbf{c}$, $\mathbf{w} = \mathbf{d}$, and expand the scalar triple products accordingly. 7. For the third identity: $$ \mathbf{a}(\mathbf{b} \cdot \mathbf{c}) + \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) + \mathbf{b}(\mathbf{c} \cdot \mathbf{a}) = \mathbf{0} $$ 8. This is a known cyclic identity involving dot products and vectors. 9. To prove it, consider the scalar triple product and properties of dot and cross products, or expand components and verify the sum is zero. 10. Each identity follows from fundamental vector algebra rules and properties of dot and cross products. Final answers: (i) $$(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c})$$ (ii) $$(\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) = [\mathbf{a}, \mathbf{b}, \mathbf{d}] \mathbf{c} - [\mathbf{a}, \mathbf{b}, \mathbf{c}] \mathbf{d}$$ (iii) $$\mathbf{a}(\mathbf{b} \cdot \mathbf{c}) + \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) + \mathbf{b}(\mathbf{c} \cdot \mathbf{a}) = \mathbf{0}$$