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The problem is asking for the **projection** and **scalar projection** of the vector **u** onto vector **v**.

We are given:
- **u** = \((-5, 1)\)
- **v** = \((-3, 2)\)

### 1. Scalar projection \(\text{scal}_{\mathbf{v}}\mathbf{u}\)

The scalar projection formula is:
\[
\text{scal}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}
\]
- **Dot product**: \(\mathbf{u} \cdot \mathbf{v} = (-5)(-3) + (1)(2) = 15 + 2 = 17\)
- **Magnitude of v**: \(|\mathbf{v}| = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}\)

Thus, the scalar projection is:
\[
\text{scal}_{\mathbf{v}}\mathbf{u} = \frac{17}{\sqrt{13}}
\]

### 2. Vector projection \(\text{proj}_{\mathbf{v}}\mathbf{u}\)

The formula for the vector projection is:
\[
\text{proj}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}
\]
- **Dot product**: \(\mathbf{v} \cdot \mathbf{v} = (-3)^2 + 2^2 = 9 + 4 = 13\)
- Projection factor: \(\frac{17}{13}\)

Thus, the vector projection is:
\[
\text{proj}_{\mathbf{v}}\mathbf{u} = \frac{17}{13} \mathbf{v} = \frac{17}{13}(-3, 2) = \left(\frac{-51}{13}, \frac{34}{13}\right)
\]

### Final answers:
- \(\text{proj}_{\mathbf{v}}\mathbf{u} = \left(\frac{-51}{13}, \frac{34}{13}\right)\)
- \(\text{scal}_{\mathbf{v}}\mathbf{u} = \frac{17}{\sqrt{13}}\)

Would you like more details on these calculations, or do you have any questions?

Here are 5 related questions:
1. How is the dot product of two vectors calculated?
2. What is the geometric meaning of vector projection?
3. How does the magnitude of a vector affect its projection?
4. Can the projection of a vector be zero? Under what conditions?
5. How does the scalar projection differ from the vector projection?

**Tip**: Always make sure to simplify radical expressions when possible to keep answers in the simplest form.

For the vectors u = (-5, 1) and v = (-3, 2), calculate proj_u v and scal_u v.

Explore the calculation of both vector and scalar projections of the vector u=(-5,1) onto the vector v=(-3,2). This tutorial explains key vector analysis concepts including the dot product and vector magnitude, with step-by-step solutions using projection formulas.

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