1. **Problem Statement:** Given vectors $A = 2\mathbf{i} - 3\mathbf{j} - \mathbf{k}$ and $B = \mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$, find: i) $C = A \times B$ ii) $D = B \times A$ iii) The magnitudes $|C|$ and $|D|$ iv) Whether $C = D$ 2. **Formula for Cross Product:** The cross product of two vectors $\mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} + u_3\mathbf{k}$ and $\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$ is given by: $$ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = (u_2 v_3 - u_3 v_2)\mathbf{i} - (u_1 v_3 - u_3 v_1)\mathbf{j} + (u_1 v_2 - u_2 v_1)\mathbf{k} $$ 3. **Calculate $C = A \times B$:** - Components of $A$: $2, -3, -1$ - Components of $B$: $1, 4, -2$ $$ C = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -3 & -1 \\ 1 & 4 & -2 \end{vmatrix} = ((-3)(-2) - (-1)(4))\mathbf{i} - (2(-2) - (-1)(1))\mathbf{j} + (2(4) - (-3)(1))\mathbf{k} $$ Simplify each component: - $\mathbf{i}$ component: $6 + 4 = 10$ - $\mathbf{j}$ component: $- ( -4 + 1 ) = -(-3) = 3$ - $\mathbf{k}$ component: $8 + 3 = 11$ So, $$ C = 10\mathbf{i} + 3\mathbf{j} + 11\mathbf{k} $$ 4. **Calculate $D = B \times A$:** Using the same formula with vectors swapped: $$ D = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 4 & -2 \\ 2 & -3 & -1 \end{vmatrix} = (4(-1) - (-2)(-3))\mathbf{i} - (1(-1) - (-2)(2))\mathbf{j} + (1(-3) - 4(2))\mathbf{k} $$ Simplify each component: - $\mathbf{i}$ component: $-4 - 6 = -10$ - $\mathbf{j}$ component: $- ( -1 - (-4) ) = -(-1 + 4) = -3$ - $\mathbf{k}$ component: $-3 - 8 = -11$ So, $$ D = -10\mathbf{i} - 3\mathbf{j} - 11\mathbf{k} $$ 5. **Calculate magnitudes $|C|$ and $|D|$:** Magnitude formula: $$ |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} $$ For $C$: $$ |C| = \sqrt{10^2 + 3^2 + 11^2} = \sqrt{100 + 9 + 121} = \sqrt{230} $$ For $D$: $$ |D| = \sqrt{(-10)^2 + (-3)^2 + (-11)^2} = \sqrt{100 + 9 + 121} = \sqrt{230} $$ 6. **Check if $C = D$:** Vectors $C$ and $D$ have the same magnitude but opposite directions: $$ D = -C $$ Therefore, $C \neq D$. **Final answers:** - $C = 10\mathbf{i} + 3\mathbf{j} + 11\mathbf{k}$ - $D = -10\mathbf{i} - 3\mathbf{j} - 11\mathbf{k}$ - $|C| = |D| = \sqrt{230}$ - $C \neq D$