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.