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.

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