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.