4. If $$ \|\vec{u}\|=13,\|\vec{v}\|=8 $$, and the direction of $$ \vec{u} $$ is $$ 55^{\circ} $$, and the direction of $$ \vec{v} $$ is $$ 155^{\circ} $$ : a. Find the component form of $$ \vec{u} $$ and $$ \vec{v} $$. Round to the nearest hundredth.

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UpStudy AI · Accepted Answer

To find the component forms of vectors $$ \vec{u} $$ and $$ \vec{v} $$, we'll use the following formulas:

$$
\vec{u} = \langle \|\vec{u}\| \cos(\theta_u), \|\vec{u}\| \sin(\theta_u) \rangle
$$
$$
\vec{v} = \langle \|\vec{v}\| \cos(\theta_v), \|\vec{v}\| \sin(\theta_v) \rangle
$$

where:
- $$ \|\vec{u}\| = 13 $$ and $$ \theta_u = 55^\circ $$
- $$ \|\vec{v}\| = 8 $$ and $$ \theta_v = 155^\circ $$

### Calculations:

1. **For $$ \vec{u} $$:**
   - $$ \cos(55^\circ) \approx 0.5736 $$
   - $$ \sin(55^\circ) \approx 0.8192 $$

$$
   u_x = 13 \times 0.5736 \approx 7.46
   $$
   $$
   u_y = 13 \times 0.8192 \approx 10.65
   $$

So, 
   $$
   \vec{u} \approx \langle 7.46, 10.65 \rangle
   $$

2. **For $$ \vec{v} $$:**
   - $$ \cos(155^\circ) \approx -0.9063 $$ (since cosine is negative in the second quadrant)
   - $$ \sin(155^\circ) \approx 0.4226 $$

$$
   v_x = 8 \times (-0.9063) \approx -7.25
   $$
   $$
   v_y = 8 \times 0.4226 \approx 3.38
   $$

So, 
   $$
   \vec{v} \approx \langle -7.25, 3.38 \rangle
   $$

### **Final Answer:**

a.  
$$
\vec{u} \approx \langle\ 7.46,\ 10.65\ \rangle \quad \text{and} \quad \vec{v} \approx \langle\ -7.25,\ 3.38\ \rangle
$$
$$ \vec{u} \approx \langle\ 7.46,\ 10.65\ \rangle \quad \text{and} \quad \vec{v} \approx \langle\ -7.25,\ 3.38\ \rangle $$

If , and the direction of is , and the direction of is :
a. Find the component form of and . Round to the nearest hundredth.

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