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