Complete parts (a) through (c) below. a. Find two unit vectors parallel to \( \mathbf{v}=-24 i-7 \mathbf{j} \). 0 of 1 b. Find \( b \) if \( v=\left(\frac{1}{6}, b\right) \) is a unit vector. c. Find all values of a such that \( \mathbf{w}=\mathrm{ai}-\frac{\mathrm{a}}{10} \mathrm{j} \) is a unit vector.

Sure, let's tackle each part step by step. --- ### **Part (a): Find Two Unit Vectors Parallel to \( \mathbf{v} = -24\,\mathbf{i} - 7\,\mathbf{j} \)** A **unit vector** is a vector with a magnitude of 1. To find unit vectors parallel to \( \mathbf{v} \), we first need to normalize \( \mathbf{v} \). 1. **Calculate the magnitude of \( \mathbf{v} \):** \[ \|\mathbf{v}\| = \sqrt{(-24)^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \] 2. **Normalize \( \mathbf{v} \) to get the first unit vector:** \[ \mathbf{u}_1 = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-24\,\mathbf{i} - 7\,\mathbf{j}}{25} = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j} \] 3. **Find the second unit vector by taking the negative of \( \mathbf{u}_1 \):** \[ \mathbf{u}_2 = -\mathbf{u}_1 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j} \] **Therefore, the two unit vectors parallel to \( \mathbf{v} \) are:** \[ \mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j} \quad \text{and} \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j} \] --- ### **Part (b): Find \( b \) if \( \mathbf{v} = \left(\frac{1}{6}, b\right) \) is a Unit Vector** For \( \mathbf{v} = \left(\frac{1}{6}, b\right) \) to be a unit vector, its magnitude must be 1. 1. **Set up the equation for the magnitude:** \[ \left\|\mathbf{v}\right\| = \sqrt{\left(\frac{1}{6}\right)^2 + b^2} = 1 \] 2. **Solve for \( b \):** \[ \sqrt{\frac{1}{36} + b^2} = 1 \\ \frac{1}{36} + b^2 = 1 \\ b^2 = 1 - \frac{1}{36} \\ b^2 = \frac{36}{36} - \frac{1}{36} \\ b^2 = \frac{35}{36} \\ b = \pm \frac{\sqrt{35}}{6} \] **Therefore, the possible values of \( b \) are:** \[ b = \frac{\sqrt{35}}{6} \quad \text{or} \quad b = -\frac{\sqrt{35}}{6} \] --- ### **Part (c): Find All Values of \( a \) Such That \( \mathbf{w} = a\,\mathbf{i} - \frac{a}{10}\,\mathbf{j} \) is a Unit Vector** For \( \mathbf{w} = a\,\mathbf{i} - \frac{a}{10}\,\mathbf{j} \) to be a unit vector, its magnitude must be 1. 1. **Express the magnitude of \( \mathbf{w} \):** \[ \left\|\mathbf{w}\right\| = \sqrt{a^2 + \left(-\frac{a}{10}\right)^2} = \sqrt{a^2 + \frac{a^2}{100}} = \sqrt{\frac{100a^2 + a^2}{100}} = \sqrt{\frac{101a^2}{100}} = \frac{\sqrt{101}\,|a|}{10} \] 2. **Set the magnitude equal to 1 and solve for \( a \):** \[ \frac{\sqrt{101}\,|a|}{10} = 1 \\ |a| = \frac{10}{\sqrt{101}} \\ a = \pm \frac{10}{\sqrt{101}} \] To rationalize the denominator: \[ a = \pm \frac{10\sqrt{101}}{101} \] **Therefore, the values of \( a \) are:** \[ a = \frac{10\sqrt{101}}{101} \quad \text{and} \quad a = -\frac{10\sqrt{101}}{101} \] --- **Summary of Answers:** - **(a)** The two unit vectors parallel to \( \mathbf{v} = -24\,\mathbf{i} -7\,\mathbf{j} \) are: \[ \mathbf{u}_1 = -\frac{24}{25}\,\mathbf{i} - \frac{7}{25}\,\mathbf{j}, \quad \mathbf{u}_2 = \frac{24}{25}\,\mathbf{i} + \frac{7}{25}\,\mathbf{j} \] - **(b)** The values of \( b \) are: \[ b = \pm \frac{\sqrt{35}}{6} \] - **(c)** The values of \( a \) are: \[ a = \pm \frac{10\sqrt{101}}{101} \]