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

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

UpStudy AI · Accepted Answer

**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.
1. $$ \mathbf{u} + \mathbf{v} = (8, 3) $$ 2. - **a.** Draw points $$ A $$, $$ B $$, and $$ C $$ in a straight line. Draw $$ \overrightarrow{AB} $$ from $$ A $$ to $$ B $$, $$ \overrightarrow{BC} $$ from $$ B $$ to $$ C $$, and $$ \overrightarrow{AC} $$ from $$ A $$ to $$ C $$. You'll see that $$ \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} $$. - **b.** Draw points $$ A $$, $$ B $$, $$ C $$, and $$ F $$ in sequence. Draw $$ \overrightarrow{AB} $$, $$ \overrightarrow{BC} $$, and $$ \overrightarrow{CF} $$. The sum $$ \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CF} $$ equals $$ \overrightarrow{AF} $$. - **c.** Draw points $$ A $$, $$ B $$, and $$ C $$. Draw $$ \overrightarrow{AB} $$ from $$ A $$ to $$ B $$, and $$ \overrightarrow{BC} $$ from $$ B $$ to $$ C $$. To represent $$ -\overrightarrow{BC} $$, draw $$ \overrightarrow{CB} $$ from $$ C $$ to $$ B $$. The sum $$ \overrightarrow{AB} + \overrightarrow{CB} $$ equals $$ \overrightarrow{AC} $$. 3. To add $$ \mathbf{u} $$ and $$ \mathbf{v} $$ graphically: - Start with $$ \mathbf{u} = (3,4) $$ from an initial point. - From the head of $$ \mathbf{u} $$, draw $$ \mathbf{v} = (5,-1) $$. - The resultant vector from the tail of $$ \mathbf{u} $$ to the head of $$ \mathbf{v} $$ is $$ (8,3) $$. These steps complete the exercise.

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