$$ \leftarrow $$ For the vectors $$ \mathbf{u}=\langle-8,0,5\rangle $$ and $$ \mathbf{v}=\langle 1,5,-5\rangle $$, calculate $$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$ and $$ \operatorname{scal}_{\mathbf{v}} \mathbf{u} $$

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To calculate the **projection** and **scalar projection** of vector **u** onto vector **v**, we'll use the following formulas:

1. **Projection of** $$ \mathbf{u} $$ **onto** $$ \mathbf{v} $$ ($$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$):
   $$
   \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v}
   $$

2. **Scalar Projection of** $$ \mathbf{u} $$ **onto** $$ \mathbf{v} $$ ($$ \operatorname{scal}_{\mathbf{v}} \mathbf{u} $$):
   $$
   \operatorname{scal}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\,\| \mathbf{v} \|\,}
   $$

Let's break down each step with the given vectors:
$$
\mathbf{u} = \langle -8, \, 0, \, 5 \rangle \quad \text{and} \quad \mathbf{v} = \langle 1, \, 5, \, -5 \rangle
$$

### 1. Calculate the Dot Products and Magnitude

**Dot Product $$ \mathbf{u} \cdot \mathbf{v} $$:**
$$
\mathbf{u} \cdot \mathbf{v} = (-8)(1) + (0)(5) + (5)(-5) = -8 + 0 - 25 = -33
$$

**Dot Product $$ \mathbf{v} \cdot \mathbf{v} $$ (which is $$ \| \mathbf{v} \|^2 $$):**
$$
\mathbf{v} \cdot \mathbf{v} = (1)^2 + (5)^2 + (-5)^2 = 1 + 25 + 25 = 51
$$

**Magnitude of $$ \mathbf{v} $$ ($$ \| \mathbf{v} \| $$):**
$$
\| \mathbf{v} \| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{51}
$$

### 2. Compute the Projection $$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$

Using the projection formula:
$$
\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{-33}{51} \right) \mathbf{v}
$$

Simplify the scalar:
$$
\frac{-33}{51} = \frac{-11}{17}
$$

Multiply by vector **v**:
$$
\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{-11}{17} \langle 1, \, 5, \, -5 \rangle = \left\langle \frac{-11}{17}, \, \frac{-55}{17}, \, \frac{55}{17} \right\rangle
$$

### 3. Compute the Scalar Projection $$ \operatorname{scal}_{\mathbf{v}} \mathbf{u} $$

Using the scalar projection formula:
$$
\operatorname{scal}_{\mathbf{v}} \mathbf{u} = \frac{-33}{\sqrt{51}}
$$

To rationalize the denominator (optional):
$$
\operatorname{scal}_{\mathbf{v}} \mathbf{u} = \frac{-33}{\sqrt{51}} \times \frac{\sqrt{51}}{\sqrt{51}} = \frac{-33 \sqrt{51}}{51} = \frac{-11 \sqrt{51}}{17}
$$

### **Final Results**

- **Projection of** $$ \mathbf{u} $$ **onto** $$ \mathbf{v} $$:
  $$
  \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \left\langle \frac{-11}{17}, \, \frac{-55}{17}, \, \frac{55}{17} \right\rangle
  $$
  
- **Scalar Projection of** $$ \mathbf{u} $$ **onto** $$ \mathbf{v} $$:
  $$
  \operatorname{scal}_{\mathbf{v}} \mathbf{u} = \frac{-33}{\sqrt{51}} \quad \text{or} \quad \frac{-11 \sqrt{51}}{17}
  $$

These results provide both the vector projection and the scalar projection of $$ \mathbf{u} $$ in the direction of $$ \mathbf{v} $$.
The projection of **u** onto **v** is $$ \left\langle \frac{-11}{17}, \frac{-55}{17}, \frac{55}{17} \right\rangle $$ and the scalar projection is $$ \frac{-33}{\sqrt{51}} $$ or $$ \frac{-11 \sqrt{51}}{17} $$.

\( \leftarrow \) For the vectors \( \mathbf{u}=\langle-8,0,5\rangle \) and \( \mathbf{v}=\langle 1,5,-5\rangle \), calculate \( \operatorname{proj}_{\mathbf{v}} \mathbf{u} \) and \( \operatorname{scal}_{\mathbf{v}} \mathbf{u} \)

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