diff --git a/docs/sph3u7.md b/docs/sph3u7.md index 0ac9314..2136be9 100644 --- a/docs/sph3u7.md +++ b/docs/sph3u7.md @@ -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 $0.005 cm$) - 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 (Source: Kognity) The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes. -$$m_{best fit} ± m_{max}-m_{min}$$ +$$m_{\text{best fit}} ± \dfrac{m_{\max}-m_{\min}}{2}$$ 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. $$|\vec{a}| = 1 \text{ m}$$ +!!! info "Reminder" + 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 (Source: Kognity) +!!! note + 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.