Let's solve each part step by step: --- ### 1. If **u** = [5, -3, -1] and **v** = [2, -1, 4]: --- #### a) \( 2\mathbf{u} - \mathbf{v} \) Calculate \( 2\mathbf{u} \): \[ 2\mathbf{u} = 2 \times [5, -3, -1] = [10, -6, -2] \] Now subtract **v**: \[ 2\mathbf{u} - \mathbf{v} = [10, -6, -2] - [2, -1, 4] = [10-2, -6-(-1), -2-4] = [8, -5, -6] \] **Final answer:** \[ \boxed{[8, -5, -6]} \] --- #### b) \( |\mathbf{u} + \mathbf{v} - 2\mathbf{k}| \) First, let's write \(\mathbf{k}\) as the unit vector in the z-direction: \(\mathbf{k} = [0, 0, 1]\). Calculate \(\mathbf{u} + \mathbf{v}\): \[ \mathbf{u} + \mathbf{v} = [5, -3, -1] + [2, -1, 4] = [7, -4, 3] \] Calculate \(2\mathbf{k}\): \[ 2\mathbf{k} = 2 \times [0, 0, 1] = [0, 0, 2] \] Now subtract: \[ \mathbf{u} + \mathbf{v} - 2\mathbf{k} = [7, -4, 3] - [0, 0, 2] = [7, -4, 1] \] Now, find the magnitude: \[ |\mathbf{u} + \mathbf{v} - 2\mathbf{k}| = \sqrt{7^2 + (-4)^2 + 1^2} = \sqrt{49 + 16 + 1} = \sqrt{66} \] **Final answer:** \[ \boxed{\sqrt{66}} \] --- #### c) Find \( G \), where \( F(3, -4, 5) \) and \( \overrightarrow{FG} = 3\mathbf{i} \) \(\overrightarrow{FG} = 3\mathbf{i} = [3, 0, 0]\) If \( F(x_1, y_1, z_1) \) and \( G(x_2, y_2, z_2) \): \[ \overrightarrow{FG} = [x_2 - x_1, y_2 - y_1, z_2 - z_1] = [3, 0, 0] \] So: \[ x_2 - 3 = 3 \implies x_2 = 6 \] \[ y_2 - (-4) = 0 \implies y_2 = -4 \] \[ z_2 - 5 = 0 \implies z_2 = 5 \] **Final answer:** \[ \boxed{G(6, -4, 5)} \] --- ### 2. If \(\mathbf{z} = [m, n, 1]\) is perpendicular to both \(\mathbf{a} = [1, -2, 2]\) and \(\mathbf{b} = [2, -3, -8]\), find \(m\) and \(n\). If \(\mathbf{z}\) is perpendicular to both, then: \[ \mathbf{z} \cdot \mathbf{a} = 0 \] \[ \mathbf{z} \cdot \mathbf{b} = 0 \] Write the dot products: 1) \( [m, n, 1] \cdot [1, -2, 2] = m \cdot 1 + n \cdot (-2) + 1 \cdot 2 = m - 2n + 2 = 0 \) 2) \( [m, n, 1] \cdot [2, -3, -8] = m \cdot 2 + n \cdot (-3) + 1 \cdot (-8) = 2m - 3n - 8 = 0 \) Now solve the system: From (1): \( m - 2n = -2 \) From (2): \( 2m - 3n = 8 \) Multiply (1) by 2: \( 2m - 4n = -4 \) Subtract from (2): \( (2m - 3n) - (2m - 4n) = 8 - (-4) \) \( 2m - 3n - 2m + 4n = 12 \) \( n = 12 \) Plug back into (1): \( m - 2(12) = -2 \implies m - 24 = -2 \implies m = 22 \) **Final answer:** \[ \boxed{m = 22, \; n = 12} \] --- Let me know if you need further clarifications!

Thank you for pointing out the correction. Let's revisit **Part a)** of question 1 with the proper calculation: --- ### 1a) Find \( 2\mathbf{v} - \mathbf{u} \) Given: \[ \mathbf{u} = [5, -3, -1] \] \[ \mathbf{v} = [2, -1, 4] \] **Step 1:** Calculate \( 2\mathbf{v} \): \[ 2 \times [2, -1, 4] = [4, -2, 8] \] **Step 2:** Subtract \(\mathbf{u}\): \[ 2\mathbf{v} - \mathbf{u} = [4, -2, 8] - [5, -3, -1] = [4 - 5, -2 - (-3), 8 - (-1)] = [-1, 1, 9] \] --- ### **Final answer:** \[ \boxed{[-1, 1, 9]} \] --- If you need further clarifications or additional parts, feel free to ask!

1a) \([-1, 1, 9]\) 1b) \(\sqrt{66}\) 1c) \(G(6, -4, 5)\) 2) \(m = 22, \quad n = 12\)