Vector Cb A20277

1. **Problem Statement:** We are given a parallelogram ABCD with vectors $\vec{a}$ and $\vec{b}$ representing sides $\overrightarrow{AB}$ and $\overrightarrow{AD}$ respectively. 2. **Goal:** Express the vector $\overrightarrow{CB}$ in terms of $\vec{a}$ and $\vec{b}$. 3. **Recall the properties of a parallelogram:** - Opposite sides are equal and parallel. - $\overrightarrow{AB} = \vec{a}$ - $\overrightarrow{AD} = \vec{b}$ - $\overrightarrow{BC} = \vec{b}$ (since $BC$ is parallel and equal to $AD$) - $\overrightarrow{DC} = \vec{a}$ (since $DC$ is parallel and equal to $AB$) 4. **Express $\overrightarrow{CB}$:** By definition, $\overrightarrow{CB} = \overrightarrow{OB} - \overrightarrow{OC}$ where $O$ is the origin. Since $\overrightarrow{OB} = \overrightarrow{OA} + \overrightarrow{AB} = \vec{a}$ (assuming $A$ at origin) and $\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{AD} + \overrightarrow{DC} = \vec{b} + \vec{a}$, Therefore, $$\overrightarrow{CB} = \vec{a} - (\vec{b} + \vec{a}) = \vec{a} - \vec{b} - \vec{a} = -\vec{b}$$ 5. **Final answer:** $$\boxed{\overrightarrow{CB} = -\vec{b}}$$ This means the vector from $C$ to $B$ is the negative of vector $\vec{b}$, which points from $A$ to $D$.