1. **Problem Statement:** Given vectors $\vec{A}$ and $\vec{B}$ with magnitudes and directions: - $\vec{A}$ has magnitude 8 and angle 270°. - $\vec{B}$ has magnitude 15 and angle 60°. Find the components of $\vec{A}$ and $\vec{B}$ and their sum $\vec{A} + \vec{B}$. Also, explain why the angle for $\vec{A}$ is 270°. 2. **Formula and Rules:** To find the components of a vector $\vec{V}$ with magnitude $V$ and angle $\theta$ (measured counterclockwise from the positive x-axis): $$ V_x = V \cos \theta $$ $$ V_y = V \sin \theta $$ Important: Angles are measured from the positive x-axis, counterclockwise. 3. **Calculate components of $\vec{A}$:** Given $V = 8$, $\theta = 270^\circ$: $$ A_x = 8 \cos 270^\circ = 8 \times 0 = 0 $$ $$ A_y = 8 \sin 270^\circ = 8 \times (-1) = -8 $$ So, $\vec{A} = 0 \hat{i} - 8 \hat{j}$. 4. **Calculate components of $\vec{B}$:** Given $V = 15$, $\theta = 60^\circ$: $$ B_x = 15 \cos 60^\circ = 15 \times 0.5 = 7.5 $$ $$ B_y = 15 \sin 60^\circ = 15 \times \frac{\sqrt{3}}{2} \approx 13 $$ So, $\vec{B} = 7.5 \hat{i} + 13 \hat{j}$. 5. **Sum of vectors $\vec{A} + \vec{B}$:** $$ (0 + 7.5) \hat{i} + (-8 + 13) \hat{j} = 7.5 \hat{i} + 5 \hat{j} $$ 6. **Why is the angle for $\vec{A}$ 270°?** The angle 270° corresponds to the direction pointing straight down along the negative y-axis. Since $\vec{A}$ points downward, its angle measured counterclockwise from the positive x-axis is 270°. This matches the vector's direction in the coordinate system. **Final answer:** $$ \vec{A} = 0 \hat{i} - 8 \hat{j}, \quad \vec{B} = 7.5 \hat{i} + 13 \hat{j}, \quad \vec{A} + \vec{B} = 7.5 \hat{i} + 5 \hat{j} $$