Let $\mathbf{v}=-4 \mathbf{i}+3 \mathbf{j}$ and $\mathbf{w}=-\mathbf{i}-3 \mathbf{j}$. Find $5 \mathbf{v}-4 \mathbf{w}$.
Solution
Step 1 ：Let \(\mathbf{v}=-4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{w}=-\mathbf{i}-3 \mathbf{j}\). We are asked to find \(5 \mathbf{v}-4 \mathbf{w}\).
Step 2 ：This is a linear combination of the vectors \(\mathbf{v}\) and \(\mathbf{w}\). To solve this, we can simply multiply each vector by its corresponding scalar and then add the results together.
Step 3 ：\(\mathbf{v} = [-4 3]\) and \(\mathbf{w} = [-1 -3]\).
Step 4 ：Multiplying each vector by its corresponding scalar, we get \(5 \mathbf{v} = [-20 15]\) and \(-4 \mathbf{w} = [4 12]\).
Step 5 ：Adding these two vectors together, we get \([-16 27]\).
Step 6 ：This corresponds to \(-16\mathbf{i} + 27\mathbf{j}\).
Step 7 ：Final Answer: \(5 \mathbf{v}-4 \mathbf{w} = \boxed{-16\mathbf{i} + 27\mathbf{j}}\)
