Let's solve each question step by step. --- ## **Question 5** Given: \(\varnothing = x^2 y z^3\) Find the **magnitude of the gradient** at \((1,1,2)\). ### **Step 1: Find the Gradient** The gradient of \(\varnothing(x, y, z)\) is \[ \nabla \varnothing = \left( \frac{\partial \varnothing}{\partial x}, \frac{\partial \varnothing}{\partial y}, \frac{\partial \varnothing}{\partial z} \right) \] Calculate each partial derivative: 1. \(\frac{\partial \varnothing}{\partial x} = 2x y z^3\) 2. \(\frac{\partial \varnothing}{\partial y} = x^2 z^3\) 3. \(\frac{\partial \varnothing}{\partial z} = 3x^2 y z^2\) ### **Step 2: Evaluate at (1, 1, 2)** Plug \(x = 1, y = 1, z = 2\): 1. \(\frac{\partial \varnothing}{\partial x} = 2(1)(1)(2^3) = 2 \times 1 \times 8 = 16\) 2. \(\frac{\partial \varnothing}{\partial y} = (1^2)(2^3) = 1 \times 8 = 8\) 3. \(\frac{\partial \varnothing}{\partial z} = 3(1^2)(1)(2^2) = 3 \times 1 \times 1 \times 4 = 12\) So, the gradient vector at (1,1,2) is \(\langle 16,\, 8,\, 12 \rangle\) ### **Step 3: Magnitude of the Gradient** \[ |\nabla \varnothing| = \sqrt{16^2 + 8^2 + 12^2} = \sqrt{256 + 64 + 144} = \sqrt{464} \] \[ \sqrt{464} \approx 21.54 \] **Final Answer for Question 5:** \[ \boxed{21.54} \] --- ## **Question 6** Given vectors: \(\mathbf{a} = [2, -3, 5]\), \(\mathbf{b} = [3, k, -2]\) Vectors are **orthogonal** if their dot product is 0. ### **Step 1: Write Dot Product** \[ \mathbf{a} \cdot \mathbf{b} = 2 \cdot 3 + (-3) \cdot k + 5 \cdot (-2) = 0 \] \[ 6 - 3k - 10 = 0 \] \[ -3k - 4 = 0 \] \[ -3k = 4 \] \[ k = -\frac{4}{3} \approx -1.33 \] **Final Answer for Question 6:** \[ \boxed{-1.33} \] --- ### **Summary** - **Q5:** \(21.54\) - **Q6:** \(-1.33\)