For the following exercise, use the vectors
u=i-3j and v=2i+3j
.

Find 2u-3v

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.

Learn how to perform vector operations using given vectors u=i-3j and v=2i+3j. This problem involves multiplying vectors, subtracting vectors, and combining vector components to find the resulting vector 2u - 3v. Suitable for advanced high school students and anyone interested in vector algebra.

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