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phys: misc fixes and improvements #9

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@ -36,6 +36,9 @@ Every SI unit can be expanded with metric prefixes.
| peta- | P | $10^{15}$ | f | femto- |
| exa- | E | $10^{18}$ | a | atto- |
!!! note
For easier memorization, notice that most of these prefixes are in multiples of $3$.
### Significant figures
- The leftmost non-zero digit is the **most significant digit**.
@ -112,7 +115,7 @@ Uncertainties are stated in the form of [value] ± [uncertainty]. A value is onl
To determine a measurement's absolute uncertainty, if:
- the instrument states its uncertainty, use that.
- an analog instrument is used, the last digit is estimated and appended to the end of the reported value. The estimated digit is uncertain by 5 at its order of magnitude.
- an analog instrument is used, the uncertainty is half of the smallest scale. (eg if a ruler can measure to $0.1cm$, the uncertainty would $0.005 cm$)
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eggy commented 2020-09-24 19:31:35 -04:00
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Examples mostly go in admonitions, so this one can be removed.

Examples mostly go in admonitions, so this one can be removed.
- a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude.
!!! example
@ -142,9 +145,9 @@ Error bars represent the uncertainty of the data, typically representing that da
<img src="/resources/images/error-slopes.png" width=700>(Source: Kognity)</img>
The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes.
$$m_{best fit} ± m_{max}-m_{min}$$
$$m_{\text{best fit}} ± \dfrac{m_{\max}-m_{\min}}{2}$$
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eggy commented 2020-09-24 19:32:07 -04:00
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MathJax sadly does not support dfrac. It’s just frac.

MathJax sadly does not support dfrac. It's just frac.
The uncertainty of the **intercepts** is the difference between the intercepts of the maximum and minimum lines.
$$intercept_{best fit} ± intercept_{max} - intercept_{min}$$
$$\text{intercept}_{\text{best fit}} ± \dfrac{\text{intercept}_{\max} - intercept_{\min}}{2}$$
## 1.3 - Vectors and scalars
@ -166,6 +169,9 @@ $$\vec{a} = (1, 1)$$
- The **magnitude** of a vector can be expressed as the absolute value of a vector.
$$|\vec{a}| = 1 \text{ m}$$
eggy commented 2020-09-24 19:57:31 -04:00
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The last “intercept” is missing \text.

The last "intercept" is missing `\text`.
!!! info "Reminder"
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eggy commented 2020-09-24 19:32:33 -04:00
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This can be moved into the description of magnitude or elsewhere instead of taking up valuable screen space in its own box.

This can be moved into the description of magnitude or elsewhere instead of taking up valuable screen space in its own box.
Remember that the **minimum magnitude** for any vector must be $`\ge 0`$
### Adding/subtracting vectors diagrammatically
1. Draw the first vector.
@ -179,6 +185,9 @@ When subtracting a vector, **negate** the vector being subtracted by giving it a
<img src="/resources/images/vector-subtract-direction.png" width=700>(Source: Kognity)</img>
!!! note
eggy commented 2020-09-24 19:34:33 -04:00
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This information seems like it is already there in the text box above the image.

This information seems like it is already there in the text box above the image.
magicalsoup commented 2020-09-24 19:59:18 -04:00
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No, I would highly argue that people do not remember this fact from looking at the diagram or the textbox. In addition, I wrote this here due to the fact that it was not clearly stated.

No, I would highly argue that people do not remember this fact from looking at the diagram or the textbox. In addition, I wrote this here due to the fact that it was not clearly stated.
Notice that when we are subtracting vectors ($\vec{a} - \vec{b}$), we have **tail** to **tail** and the difference vector has a direction from ($\vec{b} to \vec{a}$)
### Adding/subtracting vectors algebraically
Vectors can be broken up into two vectors (**"components"**) laying on the x- and y-axes via trigonometry such that the resultant of the two components is the original vector. This is especially helpful when adding larger (3+) numbers of vectors.