Given the vectors $\mathbf{u} = \mathbf{i} - 3\mathbf{j}$ and $\mathbf{v} = 2\mathbf{i} + 3\mathbf{j}$, we need to find $2\mathbf{u} - 3\mathbf{v}$.

Let's break this down step by step:

1. Multiply vector $\mathbf{u}$ by 2: $2\mathbf{u} = 2(\mathbf{i} - 3\mathbf{j}) = 2\mathbf{i} - 6\mathbf{j}$

2. Multiply vector $\mathbf{v}$ by 3: $3\mathbf{v} = 3(2\mathbf{i} + 3\mathbf{j}) = 6\mathbf{i} + 9\mathbf{j}$

3. Subtract $3\mathbf{v}$ from $2\mathbf{u}$: $2\mathbf{u} - 3\mathbf{v} = (2\mathbf{i} - 6\mathbf{j}) - (6\mathbf{i} + 9\mathbf{j})$ $= (2\mathbf{i} - 6\mathbf{i}) + (-6\mathbf{j} - 9\mathbf{j})$ $= -4\mathbf{i} - 15\mathbf{j}$

Thus, $2\mathbf{u} - 3\mathbf{v} = -4\mathbf{i} - 15\mathbf{j}$.

Would you like more details on this? If so, feel free to ask!

Here are five related questions for further exploration:

1. What is the magnitude of the resulting vector $-4\mathbf{i} - 15\mathbf{j}$?
2. How would you find the angle between vectors $\mathbf{u}$ and $\mathbf{v}$?
3. What is the dot product of vectors $\mathbf{u}$ and $\mathbf{v}$?
4. What would be the result of $3\mathbf{u} + 2\mathbf{v}$?
5. How can you express vector $\mathbf{u}$ in polar coordinates?

Tip: When performing vector operations, treat the components (i.e., the $i$ and $j$ components) separately and then combine the results at the end.