a. Find the component form of \( \vec{u} \) and \( \vec{v} \). Round to the nearest hundredth. \[ \left.\begin{array}{ll} U \rightarrow & \begin{array}{l} \cos \left(55^{\circ}\right) \approx 0.5736 \\ \\ \sin \left(55^{\circ}\right) \approx 0.8192 \end{array} \\ v_{x}=13 \cdot 0.5736 \approx 7.46 \\ v_{y}=13 \cdot 0.8192 \approx 10.65 \end{array}>\langle 7.46,10.65\rangle>\begin{array}{l} \cos \left(165^{\circ}\right) \approx-0.9063 \\ \sin \left(165^{\circ}\right) \approx 0.4226 \end{array} \quad \begin{array}{l} v_{x}=8 \cdot(-0.9063) \approx-7.25 \\ v_{y}=8 \cdot 0.4226 \approx 3.38 \end{array}\right\rangle\langle-7.25,3.38\rangle \] b. Find the component form of the resultant vector, \( \vec{u}+\vec{v} \). Round to the nearest hundredth.
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To find the component form of the resultant vector \( \vec{u} + \vec{v} \), you simply add the corresponding components of \( \vec{u} \) and \( \vec{v} \): \[ \vec{u} = \langle 7.46, 10.65 \rangle \] \[ \vec{v} = \langle -7.25, 3.38 \rangle \] Adding the components: \[ \text{Resultant } \vec{u} + \vec{v} = \langle 7.46 + (-7.25), 10.65 + 3.38 \rangle \] \[ = \langle 0.21, 14.03 \rangle \] So, the component form of the resultant vector is approximately \( \langle 0.21, 14.03 \rangle \). Now, you're set for any future vector adventures! You can visualize how these vectors interact in real-world scenarios, like forces or movements in sports or physics. Understanding vectors can help with everything from engineering designs to video game animations!