Let's solve each question step by step: --- ## **Question 1** **Given:** A = [2, -1, 1] B = [1, 2, -3] **Find:** \(|(2A + B) \times (A - 2B)|\) (magnitude of the cross product) ### **Step 1: Calculate \(2A\)** \(2A = 2 \times [2, -1, 1] = [4, -2, 2]\) ### **Step 2: Calculate \(2A + B\)** \([4, -2, 2] + [1, 2, -3] = [5, 0, -1]\) ### **Step 3: Calculate \(2B\)** \(2B = 2 \times [1, 2, -3] = [2, 4, -6]\) ### **Step 4: Calculate \(A - 2B\)** \([2, -1, 1] - [2, 4, -6] = [0, -5, 7]\) ### **Step 5: Cross Product \((2A+B) \times (A-2B)\)** Let \(U = [5, 0, -1]\), \(V = [0, -5, 7]\): \[ U \times V = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 0 & -1 \\ 0 & -5 & 7 \\ \end{vmatrix} \] \[ = \mathbf{i} (0 \cdot 7 - (-1) \cdot -5) - \mathbf{j} (5 \cdot 7 - (-1) \cdot 0) + \mathbf{k} (5 \cdot -5 - 0 \cdot 0) \] \[ = \mathbf{i} (0 - 5) - \mathbf{j} (35 - 0) + \mathbf{k} (-25 - 0) \] \[ = \mathbf{i}(-5) - \mathbf{j}(35) + \mathbf{k}(-25) \] \[ = [-5, -35, -25] \] ### **Step 6: Magnitude** \[ |[-5, -35, -25]| = \sqrt{(-5)^2 + (-35)^2 + (-25)^2} = \sqrt{25 + 1225 + 625} = \sqrt{1875} \] \[ \sqrt{1875} \approx 43.3 \] **Final Answer for Question 1:** \[ \boxed{43.3} \] --- ## **Question 2** Given: \(\mathbf{A} = xz \, \mathbf{i} - y^2 \, \mathbf{j} + 2x^2y \, \mathbf{k}\) Find the magnitude of \(\nabla \times \mathbf{A}\) at \((1, 1, 2)\). ### **Step 1: Curl Formula** \[ \nabla \times \mathbf{A} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ xz & -y^2 & 2x^2y \\ \end{vmatrix} \] ### **Step 2: Components** - **i-component:** \[ \left( \frac{\partial}{\partial y}(2x^2y) - \frac{\partial}{\partial z}(-y^2) \right) \] \[ \frac{\partial}{\partial y}(2x^2y) = 2x^2 \] \[ \frac{\partial}{\partial z}(-y^2) = 0 \] So, i-component: \(2x^2 - 0 = 2x^2\) - **j-component:** \[ -\left( \frac{\partial}{\partial x}(2x^2y) - \frac{\partial}{\partial z}(xz) \right) \] \[ \frac{\partial}{\partial x}(2x^2y) = 4xy \] \[ \frac{\partial}{\partial z}(xz) = x \] So, j-component: \(-(4xy - x) = -4xy + x\) - **k-component:** \[ \frac{\partial}{\partial x}(-y^2) - \frac{\partial}{\partial y}(xz) \] \[ \frac{\partial}{\partial x}(-y^2) = 0 \] \[ \frac{\partial}{\partial y}(xz) = 0 \] So, k-component: \(0 - 0 = 0\) So, \[ \nabla \times \mathbf{A} = [2x^2, -4xy + x, 0] \] ### **Step 3: Plug in (1, 1, 2)** - \(x=1, y=1\) - i-component: \(2(1)^2 = 2\) - j-component: \(-4(1)(1) + 1 = -4 + 1 = -3\) - k-component: \(0\) So, curl at (1,1,2) is \([2, -3, 0]\) ### **Step 4: Magnitude** \[ \sqrt{2^2 + (-3)^2 + 0^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \] **Final Answer for Question 2:** \[ \boxed{3.61} \] --- ## **Summary of Answers** **Q1:** 43.3 **Q2:** 3.61