1. **Problem:** Determine the angle made by the vector $\mathbf{V} = -10\mathbf{i} + 24\mathbf{j}$ with the positive x-axis and write the unit vector $\mathbf{n}$ in the direction of $\mathbf{V}$. 2. **Formula and rules:** - The angle $\theta$ a vector makes with the positive x-axis is given by $$\theta = \tan^{-1}\left(\frac{V_y}{V_x}\right)$$ - The unit vector in the direction of $\mathbf{V}$ is $$\mathbf{n} = \frac{\mathbf{V}}{|\mathbf{V}|}$$ where $|\mathbf{V}| = \sqrt{V_x^2 + V_y^2}$ is the magnitude of $\mathbf{V}$. 3. **Calculate the angle $\theta$:** Given $V_x = -10$, $V_y = 24$, $$\theta = \tan^{-1}\left(\frac{24}{-10}\right) = \tan^{-1}(-2.4)$$ Since $V_x$ is negative and $V_y$ positive, the vector lies in the second quadrant, so $$\theta = 180^\circ + \tan^{-1}(-2.4) = 180^\circ - 67.38^\circ = 112.62^\circ$$ 4. **Calculate the magnitude $|\mathbf{V}|$:** $$|\mathbf{V}| = \sqrt{(-10)^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26$$ 5. **Find the unit vector $\mathbf{n}$:** $$\mathbf{n} = \frac{1}{26}(-10\mathbf{i} + 24\mathbf{j}) = -\frac{10}{26}\mathbf{i} + \frac{24}{26}\mathbf{j} = -\frac{5}{13}\mathbf{i} + \frac{12}{13}\mathbf{j}$$ **Final answers:** - Angle with positive x-axis: $\boxed{112.62^\circ}$ - Unit vector: $\boxed{\mathbf{n} = -\frac{5}{13}\mathbf{i} + \frac{12}{13}\mathbf{j}}$