1. **Problem Statement:** Find the angle each vector makes with the positive x-axis. 2. **Formula:** The angle $\theta$ a vector $\vec{v} = ai + bj$ makes with the positive x-axis is given by: $$\theta = \tan^{-1}\left(\frac{b}{a}\right)$$ Important: Adjust the angle based on the quadrant where the vector lies. 3. **Vector a: $3i + 4j$** - Calculate $\theta = \tan^{-1}\left(\frac{4}{3}\right)$ - $\theta = \tan^{-1}(1.3333) \approx 53.13^\circ$ - Since $a=3 > 0$ and $b=4 > 0$, vector is in the first quadrant, so angle is $53.13^\circ$. 4. **Vector b: $6i - 8j$** - Calculate $\theta = \tan^{-1}\left(\frac{-8}{6}\right)$ - $\theta = \tan^{-1}(-1.3333) \approx -53.13^\circ$ - Since $a=6 > 0$ and $b=-8 < 0$, vector is in the fourth quadrant. - Angle with positive x-axis is $360^\circ - 53.13^\circ = 306.87^\circ$. 5. **Vector c: $5i + 12j$** - Calculate $\theta = \tan^{-1}\left(\frac{12}{5}\right)$ - $\theta = \tan^{-1}(2.4) \approx 67.38^\circ$ - Since $a=5 > 0$ and $b=12 > 0$, vector is in the first quadrant, so angle is $67.38^\circ$. **Final answers:** - a) $53.13^\circ$ - b) $306.87^\circ$ - c) $67.38^\circ$