Let's analyze each statement in turn to determine whether it is correct ("Σωστό") or incorrect ("Λάθος").

Given Statements:

1. (i) $\vec{a} = (x, y) \Rightarrow |\vec{a}| = \sqrt{x^2 + y^2}$

   • Analysis: This is correct. The magnitude of a vector $\vec{a} = (x, y)$ in two-dimensional space is indeed $|\vec{a}| = \sqrt{x^2 + y^2}$.
   • Conclusion: Correct.

2. (ii) $\vec{a} \perp \vec{\beta} \Rightarrow \det(\vec{a}, \vec{\beta}) = 0$

   • Analysis: This is incorrect. The determinant of two vectors $\vec{a}$ and $\vec{\beta}$ is zero if they are parallel, not perpendicular. For perpendicular vectors, the determinant is non-zero, as it represents the area of the parallelogram formed by $\vec{a}$ and $\vec{\beta}$.
   • Conclusion: Incorrect.

3. (iii) For the angle $\Phi$ formed by vector $\vec{a}$ with the x-axis $x'x$, it holds that $0^\circ \leq \Phi < 360^\circ$.

   • Analysis: This is correct. An angle $\Phi$ in the range $0^\circ \leq \Phi < 360^\circ$ covers all possible directions in a 2D plane, as angles repeat every $360^\circ$.
   • Conclusion: Correct.

4. (iv) Given points $A(x_1, y_1)$ and $B(x_2, y_2)$, then the vector $\overrightarrow{AB} = (x_1 - x_2, y_1 - y_2)$.

   • Analysis: This is incorrect. The vector $\overrightarrow{AB}$ should be defined as $\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1)$, as it represents the displacement from $A$ to $B$.
   • Conclusion: Incorrect.

5. (v) If $\lambda = \frac{4}{3}$, then $\lambda \vec{a} = (4, 3)$.

   • Analysis: This is correct only if $\vec{a} = (3, \frac{9}{4})$ or any vector such that scaling by $\frac{4}{3}$ results in $(4, 3)$. However, without knowing $\vec{a}$, we cannot verify this as universally correct. It is likely the intended statement assumes that $\vec{a} = (3, \frac{9}{4})$, making it correct in this context.
   • Conclusion: Correct, assuming $\vec{a} = (3, \frac{9}{4})$.

6. (vi) Given $A(2, 1)$ and $B(10, 7)$, then $\overrightarrow{AB} = 8\vec{i} + 6\vec{j}$.

   • Analysis: Calculating $\overrightarrow{AB} = (10 - 2)\vec{i} + (7 - 1)\vec{j} = 8\vec{i} + 6\vec{j}$, which is correct.
   • Conclusion: Correct.

Summary of Conclusions:

1. (i) Correct
2. (ii) Incorrect
3. (iii) Correct
4. (iv) Incorrect
5. (v) Correct, assuming $\vec{a} = (3, \frac{9}{4})$
6. (vi) Correct

Would you like more detailed explanations on any of these points?

Related Questions:

1. How do you calculate the magnitude of a vector in 3D space?
2. What is the determinant of two vectors, and what does it represent geometrically?
3. How can we determine the direction angle of a vector in a 2D coordinate system?
4. How do you find the vector between two points in 3D?
5. What are the implications of scalar multiplication on vector direction and magnitude?

Tip:

For vector calculations, keep in mind the direction and order of points, especially when finding a vector between two points (e.g., $\overrightarrow{AB}$ is calculated as $B - A$).