Vector $\overrightarrow{\mathbf{A}}$ has $x$ and $y$ components of $-8.70\mathrm{cm}$ and $15.0\mathrm{cm},$ respectively; vector $\overrightarrow{\mathbf{B}}$ has $x$ and $y$ components of$13.2\mathrm{cm}$ and $-6.60\mathrm{cm},$ respectively. If $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+3\overrightarrow{\mathbf{C}}=0$what are the components of C?

Key Concepts: Vector Components, Vector Addition Explanation: Given vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ and their components, we need to find the components of vector $\overrightarrow{\mathbf{C}}$ such that $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+3\overrightarrow{\mathbf{C}}=0$. Since the vectors are added component-wise, we can find the components of $\overrightarrow{\mathbf{C}}$ using algebraic manipulations. Step by Step Solution: Step 1. Let the $x$ and $y$ components of $\overrightarrow{\mathbf{C}}$ be $C_x$ and $C_y$ respectively. So, we know that $-8.70\mathrm{cm}-13.2\mathrm{cm}+3C_x=0$ and $15.0\mathrm{cm}+6.60\mathrm{cm}+3C_y=0$. Step 2. Solving the above two equations simultaneously, we can find the components of $\overrightarrow{\mathbf{C}}$. From the first equation, $C_x=\frac{8.70\mathrm{cm}+13.2\mathrm{cm}}{3}=7.30\mathrm{cm}$ From the second equation, $C_y=\frac{-15.0\mathrm{cm}-6.60\mathrm{cm}}{3}=-7.20\mathrm{cm}$. Step 3. So, the components of $\overrightarrow{\mathbf{C}}$ are $C_x=7.30\mathrm{cm}$ and $C_y=-7.20\mathrm{cm}$. Final Answer: The components of $\overrightarrow{\mathbf{C}}$ are $C_x=7.30\mathrm{cm}$ and $C_y=-7.20\mathrm{cm}$.