Vectors Exercise 5.4 1. Let $$ \mathbf{u}=(11,16) $$ and $$ \mathrm{v}=(-8,12) $$. Find $$ \begin{array}{llll}\begin{array}{ll} ext { a. } \mathbf{5 u} & ext { b. } \mathbf{3}(4 \mathbf{u})\end{array} \quad ext { c. }(5+\mathbf{3}) \mathbf{u} & ext { d. } \mathbf{4}(\mathbf{u}+\mathrm{v}) & ext { e. } 1 \cdot v \quad \mathrm{f} .0 \cdot \mathrm{v}\end{array} $$ 2. In the triangle $$ \mathrm{ABC}, \overrightarrow{A B} $$ represents $$ \mathbf{u} $$ and $$ \overrightarrow{B C} $$ represents v . If D is the midpoint of AB , then express $$ \overrightarrow{A C}, \overrightarrow{C A} $$ and $$ \overrightarrow{\mathrm{DC}} $$ in terms of $$ \mathbf{u} $$ and v .

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Vectors Exercise 5.4 1. Let $$ \mathbf{u}=(11,16) $$ and $$ \mathrm{v}=(-8,12) $$. Find $$ \begin{array}{llll}\begin{array}{ll}	ext { a. } \mathbf{5 u} & 	ext { b. } \mathbf{3}(4 \mathbf{u})\end{array} \quad 	ext { c. }(5+\mathbf{3}) \mathbf{u} & 	ext { d. } \mathbf{4}(\mathbf{u}+\mathrm{v}) & 	ext { e. } 1 \cdot v \quad \mathrm{f} .0 \cdot \mathrm{v}\end{array} $$ 2. In the triangle $$ \mathrm{ABC}, \overrightarrow{A B} $$ represents $$ \mathbf{u} $$ and $$ \overrightarrow{B C} $$ represents v . If D is the midpoint of AB , then express $$ \overrightarrow{A C}, \overrightarrow{C A} $$ and $$ \overrightarrow{\mathrm{DC}} $$ in terms of $$ \mathbf{u} $$ and v .

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**1. Given**  
$$
\mathbf{u} = (11,16), \quad \mathbf{v} = (-8,12)
$$

**a. Compute $$ 5\mathbf{u} $$:**  
$$
5\mathbf{u} = 5 \cdot (11,16) = (5 \cdot 11,\, 5 \cdot 16) = (55,80)
$$

**b. Compute $$ 3(4\mathbf{u}) $$:**  
First, calculate $$ 4\mathbf{u} $$:  
$$
4\mathbf{u} = 4 \cdot (11,16) = (44,64)
$$  
Then,  
$$
3(4\mathbf{u}) = 3 \cdot (44,64) = (132,192)
$$

**c. Compute $$ (5+3)\mathbf{u} $$:**  
$$
(5+3)\mathbf{u} = 8\mathbf{u} = 8 \cdot (11,16) = (88,128)
$$

**d. Compute $$ 4(\mathbf{u}+\mathbf{v}) $$:**  
First, find  
$$
\mathbf{u}+\mathbf{v} = (11 + (-8),\, 16+12) = (3,28)
$$  
Then,  
$$
4(\mathbf{u}+\mathbf{v}) = 4 \cdot (3,28) = (12,112)
$$

**e. Compute $$ 1\cdot \mathbf{v} $$:**  
$$
1\cdot \mathbf{v} = \mathbf{v} = (-8,12)
$$

**f. Compute $$ 0\cdot \mathbf{v} $$:**  
$$
0\cdot \mathbf{v} = (0,0)
$$

---

**2. In triangle $$ ABC $$, let**  
$$
\overrightarrow{AB} = \mathbf{u} \quad \text{and} \quad \overrightarrow{BC} = \mathbf{v}
$$  
Let $$ D $$ be the midpoint of $$ AB $$. Thus, the vector from $$ A $$ to $$ D $$ is  
$$
\overrightarrow{AD} = \frac{1}{2}\mathbf{u}
$$

**a. Express $$ \overrightarrow{AC} $$ in terms of $$ \mathbf{u} $$ and $$ \mathbf{v} $$:**  
Travel from $$ A $$ to $$ C $$ via $$ B $$:  
$$
\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \mathbf{u} + \mathbf{v}
$$

**b. Express $$ \overrightarrow{CA} $$ in terms of $$ \mathbf{u} $$ and $$ \mathbf{v} $$:**  
Since $$ \overrightarrow{CA} $$ is the opposite of $$ \overrightarrow{AC} $$:  
$$
\overrightarrow{CA} = -\overrightarrow{AC} = -(\mathbf{u} + \mathbf{v})
$$

**c. Express $$ \overrightarrow{DC} $$ in terms of $$ \mathbf{u} $$ and $$ \mathbf{v} $$:**  
One way to express $$ \overrightarrow{DC} $$ is by going from $$ D $$ to $$ C $$ via $$ A $$:  
$$
\overrightarrow{DC} = \overrightarrow{AC} - \overrightarrow{AD} 
$$  
Substitute the known vectors:  
$$
\overrightarrow{DC} = (\mathbf{u} + \mathbf{v}) - \frac{1}{2}\mathbf{u} = \left(1 - \frac{1}{2}\right)\mathbf{u} + \mathbf{v} = \frac{1}{2}\mathbf{u} + \mathbf{v}
$$
**1. Calculations:** - **a.** $$ 5\mathbf{u} = (55, 80) $$ - **b.** $$ 3(4\mathbf{u}) = (132, 192) $$ - **c.** $$ (5+3)\mathbf{u} = (88, 128) $$ - **d.** $$ 4(\mathbf{u}+\mathbf{v}) = (12, 112) $$ - **e.** $$ 1\cdot \mathbf{v} = (-8, 12) $$ - **f.** $$ 0\cdot \mathbf{v} = (0, 0) $$ **2. Vector Expressions in Triangle $$ ABC $$:** - **a.** $$ \overrightarrow{AC} = \mathbf{u} + \mathbf{v} $$ - **b.** $$ \overrightarrow{CA} = -(\mathbf{u} + \mathbf{v}) $$ - **c.** $$ \overrightarrow{DC} = \frac{1}{2}\mathbf{u} + \mathbf{v} $$

Vectors
Exercise 5.4

Let and . Find

In the triangle represents and represents v . If D is the midpoint of
AB , then express and in terms of and v .

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Vectors Exercise 5.4 Let and . Find In the triangle represents and represents v . If D is the midpoint of AB , then express and in terms of and v .