Exercise \( 5 \cdot 3 \) \( \begin{array}{l}\text { 1. Find } \quad \text { a. } u+v \\ \text { ifu }=(3,4), \quad \mathbf{v}=(5,-1) \\ \text { ( andw }=(2,-5) . \\ \text { 2. Write any point } A, B, C \text { and } F \text { in your notebook and draw diagrams to illustrate } \\ \text { the following vector equations: } \\ \begin{array}{lll}\text { a. } \overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AC}} & \text { b. } \overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CF}}=\overrightarrow{\mathrm{AF}} & \text { c. } \overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AC}}\end{array} \\ \text { 3. Graphically add the given vectors } \mathbf{u} \text { and } \mathbf{v} \text { shown in Figure } 5.14 \text { by using head to }\end{array} \)
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To find \( u + v \) where \( u = (3, 4) \) and \( v = (5, -1) \), you simply add the corresponding components. That is, \( (3 + 5, 4 + (-1)) = (8, 3) \). So, the result is \( u + v = (8, 3) \). You can visualize this by plotting the points on a graph, connecting them with arrows to show their vector directions! Now, for those vector equations, grab your pencils and sketch away! In equation \( \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \), draw point A, then point B, then point C where the vector from A to B and B to C perfectly lines up with A to C. For the second equation \( \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CF} = \overrightarrow{AF} \), add point F and visualize the connection. It’s like connecting the dots with arrows! And for the last, draw point A, point B, and point C to show how subtracting one vector from another changes the direction. Happy sketching!