What is open expression of a vector's magnitude?

Question

I'm 1st grade Physics student and my school started this week at 16.09.2019. I have seen 4.5 hours of lesson about vectors this week. We have seen such thing called Products of Vectors. My teacher told to class about Properties of Scalar Product with 6 items.

Properties of Scalar Product

1. $\vec{A}\cdot\vec{B}=\vec{A}\cdot\vec{B}$
2. $\vec{A}\cdot(\vec{B}+\vec{C})=\vec{A}\cdot\vec{B}+\vec{A}\cdot\vec{C}$
3. $\vec{A}\cdot\vec{A}=AA\cos(\hat{\vec{A}, \vec{A}})=A^2\Rightarrow A=\sqrt{AA}$
4. $\vec{A}\cdot\vec{B}\lt0\Rightarrow \epsilon\gt\frac{\pi}{2}$
5. $\vec{A}\cdot\vec{B}=0\Rightarrow AB\cos{\theta}, \theta=(\frac{\pi}{2}+2k\pi), (\frac{3\pi}{4}+2k\pi)$

$\vec{A}$ and $\vec{B}$ are perpendicular or one of $\vec{A}$ or $\vec{B}$ is $0$. 6. $\vec{A}\cdot\vec{X}=k, k\in\Re\Rightarrow$ What is $\vec{X}$? This question doesn't have only one right answer. There are many $\vec{X}$ vectors that ensure $\vec{A}\cdot\vec{X}=k$ equation.

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My first question is about 3rd item. Is the item correct?

My second question is about 4th question. Is $\theta\gt\frac{\pi}{2}$ or $\frac{\pi}{2}\lt\theta\lt\pi$ true?

And lastly we have seen such concept called Open Expression of A Vector's Magnitude.

$\vec{A}\cdot\vec{A}=(A_x\cdot\hat{x}+A_y\cdot\hat{y}+A_z\cdot\hat{z})\cdot(A_x\cdot\hat{x}+A_y\cdot\hat{y}+A_z\cdot\hat{z})$

$\vec{A}\cdot\vec{A}=A_x^2+A_y^2+A_z^2$

$\vec{A}\cdot\vec{A}=A^2$

$A=\left|\vec{A}\right|=\sqrt{A_x^2+A_y^2+A_z^2}$

Is the derivation and additional information correct? And also, what does Open Expression of A Vector's Magnitude mean?

Thanks!

Answer 1

By 3rd item do you mean $\vec A \cdot \vec A = A^2$? If so, yes that is correct. The dot product of 2 vectors is a scaler. The 4th equation is correct also but note that $\theta$ varies by $0 \rightarrow \pi$. I've never seen the term "open expression" used but from the equations above I think it means the obvious expressions. All the expressions look fine but #5. Again $0 \rightarrow \pi$.