Complete parts (a) through (c) below. a. Find two unit vectors parallel to \( \mathbf{v}=-24 i-7 \mathbf{j} \). b. Find \( b \) if \( v=\left\langle\frac{1}{2}, \mathrm{~b}\right\rangle \) is a unit vector. c. Find all values of a such that \( w=a i-\frac{a}{5} j \) is a unit vector. a. The parallel unit vector with the same direction is

To solve the problem, we will go through each part step by step. ### Part (a) We need to find two unit vectors parallel to \( \mathbf{v} = -24 \mathbf{i} - 7 \mathbf{j} \). 1. **Calculate the magnitude of \( \mathbf{v} \)**: \[ \|\mathbf{v}\| = \sqrt{(-24)^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \] 2. **Find the unit vector in the direction of \( \mathbf{v} \)**: The unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is given by: \[ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-24 \mathbf{i} - 7 \mathbf{j}}{25} = -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j} \] 3. **Find the unit vector in the opposite direction**: The unit vector in the opposite direction is: \[ -\mathbf{u} = \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j} \] Thus, the two unit vectors parallel to \( \mathbf{v} \) are: 1. \( -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j} \) 2. \( \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j} \) ### Part (b) We need to find \( b \) if \( \mathbf{v} = \left\langle \frac{1}{2}, b \right\rangle \) is a unit vector. 1. **Set up the equation for the magnitude of \( \mathbf{v} \)**: \[ \|\mathbf{v}\| = \sqrt{\left(\frac{1}{2}\right)^2 + b^2} = 1 \] 2. **Square both sides**: \[ \left(\frac{1}{2}\right)^2 + b^2 = 1 \] \[ \frac{1}{4} + b^2 = 1 \] 3. **Solve for \( b^2 \)**: \[ b^2 = 1 - \frac{1}{4} = \frac{3}{4} \] 4. **Find \( b \)**: \[ b = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \] Thus, the values of \( b \) are \( \frac{\sqrt{3}}{2} \) and \( -\frac{\sqrt{3}}{2} \). ### Part (c) We need to find all values of \( a \) such that \( \mathbf{w} = a \mathbf{i} - \frac{a}{5} \mathbf{j} \) is a unit vector. 1. **Set up the equation for the magnitude of \( \mathbf{w} \)**: \[ \|\mathbf{w}\| = \sqrt{a^2 + \left(-\frac{a}{5}\right)^2} = 1 \] 2. **Square both sides**: \[ a^2 + \left(-\frac{a}{5}\right)^2 = 1 \] \[ a^2 + \frac{a^2}{25} = 1 \] 3. **Combine the terms**: \[ a^2 \left(1 + \frac{1}{25}\right) = 1 \] \[ a^2 \left(\frac{25 + 1}{25}\right) = 1 \] \[ a^2 \left(\frac{26}{25}\right) = 1 \] 4. **Solve for \( a^2 \)**: \[ a^2 = \frac{25}{26} \] 5. **Find \( a \)**: \[ a = \pm \sqrt{\frac{25}{26}} = \pm \frac{5}{\sqrt{26}} = \pm \frac{5\sqrt{26}}{26} \] Thus, the values of \( a \) are \( \frac{5\sqrt{26}}{26} \) and \( -\frac{5\sqrt{26}}{26} \). ### Summary of Answers - **Part (a)**: The two unit vectors are \( -\frac{24}{25} \mathbf{i} - \frac{7}{25} \mathbf{j} \) and \( \frac{24}{25} \mathbf{i} + \frac{7}{25} \mathbf{j} \). - **Part (b)**: The values of \( b \) are \( \frac{\sqrt{3}}{2} \) and \( -\frac{\sqrt{3}}{2} \). - **Part (c)**: The values of \( a \) are \( \frac{5\sqrt{26}}{26} \) and \( -\frac{5\sqrt{26}}{26} \).