1. **Problem:** Find the cross products $\mathbf{a} \times \mathbf{b}$ and $\mathbf{b} \times \mathbf{a}$ for $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, 5, 6)$.\n\n2. **Formula:** The cross product of two vectors $\mathbf{a} = (a_1, a_2, a_3)$ and $\mathbf{b} = (b_1, b_2, b_3)$ is given by:\n$$\mathbf{a} \times \mathbf{b} = \left(a_2 b_3 - a_3 b_2,\; a_3 b_1 - a_1 b_3,\; a_1 b_2 - a_2 b_1\right)$$\n\n3. **Calculate $\mathbf{a} \times \mathbf{b}$:**\n- First component: $2 \times 6 - 3 \times 5 = 12 - 15 = -3$\n- Second component: $3 \times 4 - 1 \times 6 = 12 - 6 = 6$\n- Third component: $1 \times 5 - 2 \times 4 = 5 - 8 = -3$\n\nSo, $$\mathbf{a} \times \mathbf{b} = (-3, 6, -3)$$\n\n4. **Calculate $\mathbf{b} \times \mathbf{a}$:**\n- First component: $5 \times 3 - 6 \times 2 = 15 - 12 = 3$\n- Second component: $6 \times 1 - 4 \times 3 = 6 - 12 = -6$\n- Third component: $4 \times 2 - 5 \times 1 = 8 - 5 = 3$\n\nSo, $$\mathbf{b} \times \mathbf{a} = (3, -6, 3)$$\n\n5. **Important rule:** The cross product is anti-commutative, meaning $$\mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a})$$ which is confirmed by our results.\n\n**Final answers:**\n$$\mathbf{a} \times \mathbf{b} = (-3, 6, -3)$$\n$$\mathbf{b} \times \mathbf{a} = (3, -6, 3)$$