1. **State the problem:** We are given vectors \(\overrightarrow{AB} = \begin{pmatrix} 9 \\ -3 \end{pmatrix}\) and \(\overrightarrow{AC} = \begin{pmatrix} 18 \\ -12 \end{pmatrix}\). We need to find the magnitude of vector \(\overrightarrow{BC}\). 2. **Recall the vector relation:** The vector \(\overrightarrow{BC}\) can be found using the relation: $$\overrightarrow{BC} = \overrightarrow{AC} - \overrightarrow{AB}$$ 3. **Calculate \(\overrightarrow{BC}\):** $$\overrightarrow{BC} = \begin{pmatrix} 18 \\ -12 \end{pmatrix} - \begin{pmatrix} 9 \\ -3 \end{pmatrix} = \begin{pmatrix} 18 - 9 \\ -12 - (-3) \end{pmatrix} = \begin{pmatrix} 9 \\ -9 \end{pmatrix}$$ 4. **Find the magnitude \(|\overrightarrow{BC}|\):** The magnitude of a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is given by: $$|\overrightarrow{v}| = \sqrt{x^2 + y^2}$$ Applying this to \(\overrightarrow{BC}\): $$|\overrightarrow{BC}| = \sqrt{9^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162}$$ 5. **Simplify the surd:** $$\sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2}$$ **Final answer:** $$|\overrightarrow{BC}| = 9\sqrt{2}$$