Given vectors u, equals, open angle bracket, minus, 5, comma, minus, 2, close angle bracket**u**=⟨−5,−2⟩ and v, equals, open angle bracket, 4, comma, minus, 1, close angle bracket, comma**v**=⟨4,−1⟩, find the difference v, minus, u**v**−**u** and represent the difference graphically in two different ways.

Components of Difference: open angle bracket

To find the difference \( \mathbf{v} - \mathbf{u} \) between the vectors \( \mathbf{v} = \langle 4, -1 \rangle \) and \( \mathbf{u} = \langle -5, -2 \rangle \), you subtract the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \):

\[
\mathbf{v} - \mathbf{u} = \langle 4, -1 \rangle - \langle -5, -2 \rangle
\]

Now, perform the subtraction:

\[
\mathbf{v} - \mathbf{u} = \langle 4 - (-5), -1 - (-2) \rangle
\]
\[
\mathbf{v} - \mathbf{u} = \langle 4 + 5, -1 + 2 \rangle
\]
\[
\mathbf{v} - \mathbf{u} = \langle 9, 1 \rangle
\]

So, the difference vector \( \mathbf{v} - \mathbf{u} \) is \( \langle 9, 1 \rangle \).

### Graphical Representation

You can represent the difference vector \( \mathbf{v} - \mathbf{u} \) graphically in two different ways:

1. **Tip-to-Tail Method:**
   - First, plot the vectors \( \mathbf{u} \) and \( \mathbf{v} \) starting from the origin.
   - Then, plot the vector \( \mathbf{u} \) starting from the origin and the vector \( \mathbf{v} \) starting from the tip of \( \mathbf{u} \).
   - The vector \( \mathbf{v} - \mathbf{u} \) is the vector that starts from the tail of \( \mathbf{u} \) and ends at the tip of \( \mathbf{v} \).

2. **Parallelogram Method:**
   - Plot both vectors \( \mathbf{u} \) and \( \mathbf{v} \) starting from the origin.
   - Draw a parallelogram using \( \mathbf{u} \) and \( \mathbf{v} \) as adjacent sides.
   - The diagonal of the parallelogram from the origin represents the vector \( \mathbf{v} - \mathbf{u} \).

Would you like more details or have any questions about this solution? Here are some related questions:

1. How do you graphically represent the sum of two vectors?
2. What is the geometric interpretation of vector subtraction?
3. Can you find the magnitude of the difference vector \( \mathbf{v} - \mathbf{u} \)?
4. How do you find the unit vector in the direction of \( \mathbf{v} - \mathbf{u} \)?
5. What are the properties of vector addition and subtraction in general?

**Tip:** Always double-check the signs when subtracting vectors to avoid common mistakes!

Learn how to find the difference vector v - u between vectors v = ⟨4, -1⟩ and u = ⟨-5, -2⟩. Includes graphical representations using the tip-to-tail and parallelogram methods. Suitable for high school students studying vectors in mathematics.

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