Given vectors u, equals, open angle bracket, 3, comma, 7, close angle bracket**u**=⟨3,7⟩ and v, equals, open angle bracket, 7, comma, minus, 3, close angle bracket, comma**v**=⟨7,−3⟩, find the difference v, minus, u**v**−**u** and represent the difference graphically in two different ways.

Components of Difference:

To find the difference \( \mathbf{v} - \mathbf{u} \) between the vectors \( \mathbf{v} = \langle 7, -3 \rangle \) and \( \mathbf{u} = \langle 3, 7 \rangle \), you subtract the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \):

\[
\mathbf{v} - \mathbf{u} = \langle 7, -3 \rangle - \langle 3, 7 \rangle
\]

Now, perform the subtraction:

\[
\mathbf{v} - \mathbf{u} = \langle 7 - 3, -3 - 7 \rangle
\]
\[
\mathbf{v} - \mathbf{u} = \langle 4, -10 \rangle
\]

So, the difference vector \( \mathbf{v} - \mathbf{u} \) is \( \langle 4, -10 \rangle \).

### 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 \( \mathbf{u} = \langle 3, 7 \rangle \) and \( \mathbf{v} = \langle 7, -3 \rangle \) 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 \( \mathbf{v} - \mathbf{u} = \langle 4, -10 \rangle \) 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 \( \mathbf{v} - \mathbf{u} = \langle 4, -10 \rangle \).

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

1. How do you compute the sum of two vectors?
2. What is the geometric significance of the vector \( \mathbf{v} - \mathbf{u} \)?
3. Can you calculate the magnitude of the vector \( \mathbf{v} - \mathbf{u} \)?
4. How can you find a vector perpendicular to \( \mathbf{v} - \mathbf{u} \)?
5. What is the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \)?

**Tip:** Drawing vectors on a coordinate plane helps visualize their sum, difference, and other operations!

Learn how to find the difference between vectors v and u, represented as v - u = ⟨4, -10⟩. This problem covers vector subtraction and graphical representation methods such as tip-to-tail and parallelogram methods. Suitable for high school students studying vectors in mathematics.

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