1. **State the problem:** Calculate the vector expression $$3\mathbf{a} - \frac{1}{4}(\mathbf{c} - \mathbf{b})$$ where $$\mathbf{a} = \begin{pmatrix}-8 \\ 7\end{pmatrix}, \mathbf{b} = \begin{pmatrix}8 \\ -20\end{pmatrix}, \mathbf{c} = \begin{pmatrix}0 \\ -16\end{pmatrix}$$. 2. **Recall vector operations:** - Scalar multiplication: multiply each component by the scalar. - Vector subtraction: subtract corresponding components. - Distributive property applies. 3. **Calculate each part:** - Multiply $$\mathbf{a}$$ by 3: $$3\mathbf{a} = 3 \times \begin{pmatrix}-8 \\ 7\end{pmatrix} = \begin{pmatrix}3 \times -8 \\ 3 \times 7\end{pmatrix} = \begin{pmatrix}-24 \\ 21\end{pmatrix}$$ - Calculate $$\mathbf{c} - \mathbf{b}$$: $$\mathbf{c} - \mathbf{b} = \begin{pmatrix}0 \\ -16\end{pmatrix} - \begin{pmatrix}8 \\ -20\end{pmatrix} = \begin{pmatrix}0 - 8 \\ -16 - (-20)\end{pmatrix} = \begin{pmatrix}-8 \\ 4\end{pmatrix}$$ - Multiply $$\frac{1}{4} (\mathbf{c} - \mathbf{b})$$: $$\frac{1}{4} \times \begin{pmatrix}-8 \\ 4\end{pmatrix} = \begin{pmatrix}\frac{1}{4} \times -8 \\ \frac{1}{4} \times 4\end{pmatrix} = \begin{pmatrix}-2 \\ 1\end{pmatrix}$$ 4. **Combine the results:** $$3\mathbf{a} - \frac{1}{4}(\mathbf{c} - \mathbf{b}) = \begin{pmatrix}-24 \\ 21\end{pmatrix} - \begin{pmatrix}-2 \\ 1\end{pmatrix} = \begin{pmatrix}-24 - (-2) \\ 21 - 1\end{pmatrix} = \begin{pmatrix}-22 \\ 20\end{pmatrix}$$ **Final answer:** $$\boxed{\begin{pmatrix}-22 \\ 20\end{pmatrix}}$$