Exercise \( 5 \cdot 3 \) \( \begin{array}{l}\text { 1. Find } \quad \text { a. } u+v \\ \text { ifu }=(3,4), \quad \mathbf{v}=(5,-1) \\ \text { ( andw }=(2,-5) . \\ \text { 2. Write any point } A, B, C \text { and } F \text { in your notebook and draw diagrams to illustrate } \\ \text { the following vector equations: } \\ \begin{array}{lll}\text { a. } \overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AC}} & \text { b. } \overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CF}}=\overrightarrow{\mathrm{AF}} & \text { c. } \overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AC}}\end{array} \\ \text { 3. Graphically add the given vectors } \mathbf{u} \text { and } \mathbf{v} \text { shown in Figure } 5.14 \text { by using head to }\end{array} \)

**1. Compute \( \mathbf{u} + \mathbf{v} \) where \( \mathbf{u} = (3,4) \) and \( \mathbf{v} = (5,-1) \):** - To add two vectors, add their corresponding components. - Compute the first component: \( 3 + 5 = 8 \). - Compute the second component: \( 4 + (-1) = 3 \). Thus, \[ \mathbf{u} + \mathbf{v} = (8,3). \] **2. Illustrate the vector equations using any points \( A \), \( B \), \( C \), and \( F \):** - **(a) \( \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \):** Choose three points along a path, for example, let \( A \), \( B \), and \( C \) lie on a line (or form a broken line): - Draw point \( A \). - Draw point \( B \) and draw the vector \( \overrightarrow{AB} \) from \( A \) to \( B \). - Draw point \( C \) and draw the vector \( \overrightarrow{BC} \) from \( B \) to \( C \). The vector from \( A \) directly to \( C \) is \( \overrightarrow{AC} \), which is the sum \( \overrightarrow{AB} + \overrightarrow{BC} \). - **(b) \( \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CF} = \overrightarrow{AF} \):** Choose four points in sequence: - Draw point \( A \). - Draw point \( B \) and vector \( \overrightarrow{AB} \). - Draw point \( C \) and vector \( \overrightarrow{BC} \) starting at \( B \). - Draw point \( F \) and vector \( \overrightarrow{CF} \) starting at \( C \). The sum of vectors \( \overrightarrow{AB} \), \( \overrightarrow{BC} \), and \( \overrightarrow{CF} \) gives the resultant vector from \( A \) to \( F \), which is \( \overrightarrow{AF} \). - **(c) \( \overrightarrow{AB} - \overrightarrow{BC} = \overrightarrow{AC} \):** To illustrate subtraction, you can do the following: - Draw points \( A \), \( B \), and \( C \). - Draw \( \overrightarrow{AB} \) from \( A \) to \( B \). - To depict \( -\overrightarrow{BC} \), draw the vector \( \overrightarrow{CB} \) (which has the same magnitude as \( \overrightarrow{BC} \) but points from \( C \) to \( B \)). The sum \( \overrightarrow{AB} + \overrightarrow{CB} \) gives the vector \( \overrightarrow{AC} \) from \( A \) to \( C \). **3. Graphically add the vectors \( \mathbf{u} \) and \( \mathbf{v} \):** - Draw the vector \( \mathbf{u} = (3,4) \) starting from an initial point (you can label this as the tail). - From the head of \( \mathbf{u} \), draw the vector \( \mathbf{v} = (5,-1) \) (using the head-to-tail method). - The resulting vector from the tail of \( \mathbf{u} \) to the head of \( \mathbf{v} \) represents \( \mathbf{u} + \mathbf{v} \). - As computed earlier, this resultant vector is \( (8,3) \). These steps complete the solution of the exercise.