The image you uploaded contains two mathematical problems written in Khmer with complex numbers and vectors. I'll break down the contents and provide a detailed explanation of each part.

Problem 1 (Question 25):

This problem involves complex numbers:

You are given: $Z = \frac{1+i}{(2+i)i}$ and $W = (2+i)x + y$

You are asked to:

• Solve for $Z$.
• Use the result of $Z$ to calculate a certain expression involving $Z$ and $W$.

Steps to approach:

1. Simplify $Z$:

   • First, simplify the denominator $(2+i)i$.
   • Then simplify the fraction to get $Z$ in the form $a+bi$.

2. Substitute and simplify $W$:

   • It looks like $x$ and $y$ are variables, and possibly, once $Z$ is solved, you'll use it to find a relation between $x$, $y$, or further expressions.

Problem 2 (Question 30):

This problem is about vectors:

You are given the following conditions for vectors $\vec{u}$, $\vec{v}$, and their properties:

1. $|\vec{u}| = 10$ and $\vec{u} \cdot \vec{v} = 7 \cos \frac{\pi}{4}$, find the magnitude of $\vec{v}$.
2. $|\vec{v}| = 2$, $\cos \frac{\pi}{3}$, calculate $\vec{u} \cdot \vec{v}$.
3. $|\vec{u}| = 2\sqrt{2}$, $|\vec{v}| = 2$, find $\cos \alpha$, where $\alpha$ is the angle between $\vec{u}$ and $\vec{v}$.

Steps to approach:

1. For each part of the problem, use vector dot product formulas:
   • $\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta$.
   • Given magnitudes and angles, compute the dot product or the required vector magnitude.

Would you like a detailed solution of any of these problems, or do you have any specific questions about the content?

5 Related Questions:

1. How do you simplify complex numbers like $\frac{1+i}{(2+i)i}$?
2. How can you represent vectors geometrically in two-dimensional space?
3. What is the significance of the dot product in vector operations?
4. How do you find the angle between two vectors using the dot product formula?
5. What are the applications of complex numbers in electrical engineering?

Tip: When simplifying complex fractions, always multiply by the conjugate of the denominator to eliminate imaginary numbers from the denominator.