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Let's solve each problem one by one.

### Problem 1:
In \(\mathbb{R}^2\), find the equation of the line symmetric to \(y = x + 2\) with respect to the line \(y = 2x\).

To find the equation of the line symmetric to \(y = x + 2\) with respect to \(y = 2x\):
1. Convert \(y = x + 2\) and \(y = 2x\) into parametric forms.
2. Reflect each point on the line \(y = x + 2\) over the line \(y = 2x\).
3. Derive the new line's equation.

**Solution:**

The line \(y = x + 2\) can be written parametrically as:
\[
(x, x + 2)
\]

For reflection across \(y = 2x\), we use the following transformation for a point \((a, b)\) reflected over \(y = kx\):
\[
(x', y') = \left( \frac{a + k(b - ka)}{1 + k^2}, \frac{ka + b - k^2a}{1 + k^2} \right)
\]

For \(y = 2x\) (i.e., \(k = 2\)):
\[
(x', y') = \left( \frac{x + 2(y - 2x)}{1 + 2^2}, \frac{2x + y - 2^2x}{1 + 2^2} \right)
= \left( \frac{x + 2y - 4x}{5}, \frac{2x + y - 4x}{5} \right)
= \left( \frac{-3x + 2y}{5}, \frac{-2x + y}{5} \right)
\]

Thus, substituting \(y = x + 2\):
\[
x' = \frac{-3x + 2(x + 2)}{5} = \frac{-3x + 2x + 4}{5} = \frac{-x + 4}{5}
\]
\[
y' = \frac{-2x + (x + 2)}{5} = \frac{-2x + x + 2}{5} = \frac{-x + 2}{5}
\]

The new line can be written as \(y' = x' + \frac{2}{5}\).

### Problem 2:
In \(\mathbb{R}^2\), using the point \(\left(0, -\frac{\sqrt{2}}{3}\right)\) as a base point, find the equation of the curve \(x^2 + 3y^2 + 2\sqrt{2}y = 0\) magnified \(\sqrt{3}\) times along the y-axis.

**Solution:**

Given curve:
\[
x^2 + 3y^2 + 2\sqrt{2}y = 0
\]

Magnifying \(y\)-axis by \(\sqrt{3}\) means replacing \(y\) with \(\frac{y}{\sqrt{3}}\):
\[
x^2 + 3\left(\frac{y}{\sqrt{3}}\right)^2 + 2\sqrt{2}\left(\frac{y}{\sqrt{3}}\right) = 0
\]
\[
x^2 + y^2 + \frac{2\sqrt{2}y}{\sqrt{3}} = 0
\]
\[
x^2 + y^2 + \frac{2\sqrt{2}}{\sqrt{3}}y = 0
\]
\[
x^2 + y^2 + \frac{2\sqrt{2}}{\sqrt{3}}y = 0
\]

### Problem 3:
In \(\mathbb{R}^3\), find the coordinates of the point (4,5,6) after a \(\frac{\pi}{2}\) rotation about the line \(x = \frac{y}{2} = \frac{z}{3}\).

**Solution:**

The axis of rotation can be parameterized as:
\[
(x, 2x, 3x)
\]

Rotation of a point in 3D about a line requires complex matrix operations or quaternions.

### Problem 4:
In \(\mathbb{R}^3\), find the equation of the line symmetric to \(x = \frac{y}{2} = \frac{z}{3}\) with respect to the plane \(x + 2y + 3z = 4\).

**Solution:**

Reflection of a line

Explore how to solve problems involving reflections of lines using transformation geometry. This detailed solution covers parametric equations, point reflection formulas, and applications in advanced mathematics.

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