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The task involves solving part **2a** based on the provided example.

### The first image:
The equation is:
\[
\cos(\alpha) = \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{\|\mathbf{v}_1\| \|\mathbf{v}_2\|} = \frac{6}{\sqrt{6} \cdot \sqrt{11}}
\]
where:
\[
\mathbf{v}_1 = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -3 \\ 1 \\ 1 \end{pmatrix}
\]
This calculates the cosine of the angle between two vectors.

### The second image:
It gives a pair of parametric plane equations.

For **2a**, let's extract the vectors and compute the angle between the normal vectors of two planes.

The parametric plane equations are given as:
- \( E_1: \left( \mathbf{x} - \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} \right) \cdot \begin{pmatrix} 5 \\ 0 \\ 1 \end{pmatrix} = 0 \)
- \( E_2: \left( \mathbf{x} - \begin{pmatrix} 2 \\ 3 \\ 7 \end{pmatrix} \right) \cdot \begin{pmatrix} 6 \\ 1 \\ 0 \end{pmatrix} = 0 \)

To solve this:
1. Extract the normal vectors from the plane equations.
   - Normal vector to \( E_1 \): \( \mathbf{n}_1 = \begin{pmatrix} 5 \\ 0 \\ 1 \end{pmatrix} \)
   - Normal vector to \( E_2 \): \( \mathbf{n}_2 = \begin{pmatrix} 6 \\ 1 \\ 0 \end{pmatrix} \)

2. Use the dot product formula to find the angle \( \alpha \) between the two vectors:
   \[
   \cos(\alpha) = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}
   \]
   Let's compute it:

   - Dot product \( \mathbf{n}_1 \cdot \mathbf{n}_2 = (5 \cdot 6) + (0 \cdot 1) + (1 \cdot 0) = 30 \)
   - Magnitudes:
     \[
     \|\mathbf{n}_1\| = \sqrt{5^2 + 0^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}
     \]
     \[
     \|\mathbf{n}_2\| = \sqrt{6^2 + 1^2 + 0^2} = \sqrt{36 + 1} = \sqrt{37}
     \]

3. Plugging these values into the formula:
   \[
   \cos(\alpha) = \frac{30}{\sqrt{26} \cdot \sqrt{37}} = \frac{30}{\sqrt{962}} \approx 0.967
   \]
4. Find the angle \( \alpha \) using the inverse cosine:
   \[
   \alpha \approx \cos^{-1}(0.967) \approx 15.05^\circ
   \]

Thus, the angle between the two planes is approximately \( 15.05^\circ \).

---

Do you need more details on this process, or should I proceed with any further steps?

### 5 Questions for Further Exploration:
1. How are dot products related to the angle between vectors?
2. What is the geometric meaning of a normal vector to a plane?
3. Can you solve the magnitude of vectors geometrically without using the formula?
4. How does this concept extend to higher-dimensional spaces (beyond 3D)?
5. What happens if the normal vectors are orthogonal? What is the angle between the planes?

### Tip:
When computing angles between vectors, always ensure your vectors are in their simplest form and normalized if necessary.

This page explains how to solve part 2a of an example using vector algebra. It demonstrates how to compute the cosine of the angle between two vectors using the dot product formula and provides a detailed breakdown of the steps.

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