1. **Problem Statement:** We want to understand how to add two vectors together and find the resultant vector. 2. **Formula and Rules:** Vector addition is done by adding corresponding components of the vectors. If \( \vec{A} = (A_x, A_y) \) and \( \vec{B} = (B_x, B_y) \), then their sum \( \vec{R} = \vec{A} + \vec{B} = (A_x + B_x, A_y + B_y) \). 3. **Example:** Let \( \vec{A} = (3, 4) \) and \( \vec{B} = (1, 2) \). 4. **Step-by-step Addition:** - Add the x-components: \( 3 + 1 = 4 \) - Add the y-components: \( 4 + 2 = 6 \) 5. **Resultant Vector:** \[ \vec{R} = (4, 6) \] 6. **Explanation:** This means if you move 3 units right and 4 units up, then from that point move 1 unit right and 2 units up, your total movement is 4 units right and 6 units up. 7. **Magnitude of Resultant Vector:** \[ |\vec{R}| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \approx 7.21 \] This shows the length of the resultant vector. Thus, vector addition combines the directions and magnitudes component-wise to give a new vector.