From d104d808948ece9b6b9be03c921df7eccda70314 Mon Sep 17 00:00:00 2001 From: eggy Date: Mon, 21 Sep 2020 19:44:47 -0400 Subject: [PATCH 1/5] math: add frequency distribution --- docs/mhf4u7.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/docs/mhf4u7.md b/docs/mhf4u7.md index 08907e4..ea05953 100644 --- a/docs/mhf4u7.md +++ b/docs/mhf4u7.md @@ -2,6 +2,18 @@ The course code for this page is **MHF4U7**. +## 4 - Statistics and probability + +!!! note "Definition" + - **Descriptive statistics:** The use of methods to organise, display, and describe data by using various charts and summary methods to reduce data to a manageable size. + - **Inferential statistics:** The use of samples to make judgements about a population. + - **Data set:** A collection of data with elements and observations, typically in the form of a table. It is similar to a map or dictionary in programming. + - **Element:** The name of an observation(s), similar to a key to a map/dictionary in programming. + - **Observation:** The collected data linked to an element, similar to a value to a map/dictionary in programming. + - **Raw data:** Data collected prior to processing or ranking. + +### Frequency distribution + ## Resources From 9185cd95031fb5f511dab875aa54f67c7ea2cde0 Mon Sep 17 00:00:00 2001 From: eggy Date: Mon, 21 Sep 2020 20:35:15 -0400 Subject: [PATCH 2/5] phys: add kognity vector information --- docs/sph3u7.md | 62 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 62 insertions(+) diff --git a/docs/sph3u7.md b/docs/sph3u7.md index b4ba6ae..61cb5b8 100644 --- a/docs/sph3u7.md +++ b/docs/sph3u7.md @@ -149,6 +149,68 @@ $$intercept_{best fit} ± intercept_{max} - intercept_{min}$$ ## 1.3 - Vectors and scalars +!!! note "Definition" + - **Scalar:** A physical quantity with a numerical value and unit. + - **Vector:** A physical quantity with a numerical value, unit, and **direction.** + +??? example + - Physical quantities represented by scalars include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more. + - Physical quantities represented by vectors include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more. + +Vectors are represented as arrows whose length represents their scale/magnitude and their orientation refer to their direction. + +### Adding/subtracting vectors diagrammatically + +1. Draw the first vector. +2. Draw the second vector with its *tail* at the *head* of the first vector. +3. Repeat step 2 as necessary for as many vectors as you want by attaching them to the *head* of the last vector. +4. Draw a new vector from the *tail* of the first vector to the *head* of the last vector. + +(Source: Kognity) + +When subtracting a vector, **negate** the vector being subtracted by giving it an opposite direction. + +(Source: Kognity) + +### Parallelogram rule + +The parallelogram rule states that the sum of two vectors that form two sides of a parallelogram is the diagonal of that parallelogram. + +(Source: Kognity) + +### Multiplying vectors and scalars + +The product of a vector multiplied by a scalar is a vector with a magnitude of the vector multiplied by the scalar with the same direction as the original vector. + +$$v[direction] × s = (v×s)[direction]$$ + +### Vector decomposition + +By breaking up a vector into lengths along the x- and y-axes, the sum of two vectors can be calculated algebraically. + +(Source: Kognity) + +For vector $\textbf{a}$ and vector $\textbf{b}$: + +$$ +a_{x} = a\cos\theta_{a} \\ +a_{y} = a\sin\theta_{a} +$$ + +Proof: +$a=\sqrt{a^{2}_{x}+a^{2}_{y}} \\$ +$=\sqrt{(a\cos\theta_{a})^2 + (a\sin\theta{a})^2} \\$ +$=\sqrt{a^2(\cos\theta_{a}^2 + \sin\theta_{a}^2)} \\$ +$=\sqrt{a^2} \\$ +$=a$ + +From the diagram above, we can figure out that: + +$$ +r_{magnitude}=\sqrt{(a\cos\theta_{a} + b\cos\theta_{b})^2 + (a\sin\theta_{a} + b\cos\theta_{b})^2} \\ +r_{direction}=\tan^{-1}(\frac{a\sin\theta_{a} + b\sin\theta_{b}}{a\cos\theta_{a} + b\cos\theta_{b}}) +$$ + ## Resources - [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf) From eab75db11a63669c1de638af08ce2a2b1456d151 Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 22 Sep 2020 09:35:06 -0400 Subject: [PATCH 3/5] phys: add more introductory information on vectors --- docs/sph3u7.md | 21 ++++++++++++++++----- 1 file changed, 16 insertions(+), 5 deletions(-) diff --git a/docs/sph3u7.md b/docs/sph3u7.md index 61cb5b8..5fdc1da 100644 --- a/docs/sph3u7.md +++ b/docs/sph3u7.md @@ -150,14 +150,25 @@ $$intercept_{best fit} ± intercept_{max} - intercept_{min}$$ ## 1.3 - Vectors and scalars !!! note "Definition" - - **Scalar:** A physical quantity with a numerical value and unit. - - **Vector:** A physical quantity with a numerical value, unit, and **direction.** + - **Scalar:** A physical quantity with a numerical value (magnitude) and a unit. + - **Vector:** A physical quantity with a numerical value (magnitude), a unit, and a **direction.** ??? example - - Physical quantities represented by scalars include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more. - - Physical quantities represented by vectors include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more. + - Scalar quantities include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more. + - Vector quantities include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more. -Vectors are represented as arrows whose length represents their scale/magnitude and their orientation refer to their direction. +Vectors are drawn as arrows whose length represents their scale/magnitude and their orientation refer to their direction. + +!!! info + - A variable representing a vector is written with a right-pointing arrow above it. + - The **standard form** of a vector is expressed as its magnitude followed by its unit followed by its direction in square brackets. + $$\vec{a} = 1\text{ m }[N 45° E]$$ + - The **component form** of a vector is expressed as the location of its head on a cartesian plane if its tail were at $(0, 0)$. + $$\vec{a} = (1, 1)$$ + +!!! example + The below shows a vector with a magnitude of 1, unit of metre, pointing north. + $$\vec{a} = 1\text{ m }[N]$$ ### Adding/subtracting vectors diagrammatically From 4393424e4c8c43c73a8ec71f84f1272d267194fe Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 22 Sep 2020 09:58:39 -0400 Subject: [PATCH 4/5] phys: convert vectors to use vector notation --- docs/sph3u7.md | 42 +++++++++++++++++++----------------------- 1 file changed, 19 insertions(+), 23 deletions(-) diff --git a/docs/sph3u7.md b/docs/sph3u7.md index 5fdc1da..2ca6465 100644 --- a/docs/sph3u7.md +++ b/docs/sph3u7.md @@ -157,29 +157,25 @@ $$intercept_{best fit} ± intercept_{max} - intercept_{min}$$ - Scalar quantities include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more. - Vector quantities include velocity, displacement, acceleration, force (e.g., weight), momentum, impulse, and more. -Vectors are drawn as arrows whose length represents their scale/magnitude and their orientation refer to their direction. +Vectors are drawn as arrows whose length represents their scale/magnitude and their orientation refer to their direction. A variable representing a vector is written with a right-pointing arrow above it. -!!! info - - A variable representing a vector is written with a right-pointing arrow above it. - - The **standard form** of a vector is expressed as its magnitude followed by its unit followed by its direction in square brackets. - $$\vec{a} = 1\text{ m }[N 45° E]$$ - - The **component form** of a vector is expressed as the location of its head on a cartesian plane if its tail were at $(0, 0)$. - $$\vec{a} = (1, 1)$$ - -!!! example - The below shows a vector with a magnitude of 1, unit of metre, pointing north. - $$\vec{a} = 1\text{ m }[N]$$ + - The **standard form** of a vector is expressed as its magnitude followed by its unit followed by its direction in square brackets. +$$\vec{a} = 1\text{ m }[N 45° E]$$ + - The **component form** of a vector is expressed as the location of its head on a cartesian plane if its tail were at $(0, 0)$. +$$\vec{a} = (1, 1)$$ + - The **magnitude** of a vector can be expressed as the absolute value of a vector. +$$|\vec{a}| = 1 \text{ m}$$ ### Adding/subtracting vectors diagrammatically 1. Draw the first vector. 2. Draw the second vector with its *tail* at the *head* of the first vector. 3. Repeat step 2 as necessary for as many vectors as you want by attaching them to the *head* of the last vector. -4. Draw a new vector from the *tail* of the first vector to the *head* of the last vector. +4. Draw a new (**resultant**) vector from the *tail* of the first vector to the *head* of the last vector. (Source: Kognity) -When subtracting a vector, **negate** the vector being subtracted by giving it an opposite direction. +When subtracting a vector, **negate** the vector being subtracted by giving it an opposite direction and then add the vectors. (Source: Kognity) @@ -201,25 +197,25 @@ By breaking up a vector into lengths along the x- and y-axes, the sum of two vec (Source: Kognity) -For vector $\textbf{a}$ and vector $\textbf{b}$: +For vector $\vec{a}$ and vector $\vec{b}$: $$ -a_{x} = a\cos\theta_{a} \\ -a_{y} = a\sin\theta_{a} +\vec{a}_{x} = a\cos\theta_{a} \\ +\vec{a}_{y} = a\sin\theta_{a} $$ Proof: -$a=\sqrt{a^{2}_{x}+a^{2}_{y}} \\$ -$=\sqrt{(a\cos\theta_{a})^2 + (a\sin\theta{a})^2} \\$ -$=\sqrt{a^2(\cos\theta_{a}^2 + \sin\theta_{a}^2)} \\$ -$=\sqrt{a^2} \\$ -$=a$ +$|\vec{a}| = \sqrt{a^{2}_{x}+a^{2}_{y}} \\$ +$= \sqrt{(|\vec{a}|\cos\theta_{a})^2 + (|\vec{a}|\sin\theta{a})^2} \\$ +$=\sqrt{|\vec{a}|^2(\cos\theta_{a}^2 + \sin\theta_{a}^2)} \\$ +$=\sqrt{|\vec{a}|^2} \\$ +$=|\vec{a}|$ From the diagram above, we can figure out that: $$ -r_{magnitude}=\sqrt{(a\cos\theta_{a} + b\cos\theta_{b})^2 + (a\sin\theta_{a} + b\cos\theta_{b})^2} \\ -r_{direction}=\tan^{-1}(\frac{a\sin\theta_{a} + b\sin\theta_{b}}{a\cos\theta_{a} + b\cos\theta_{b}}) +|\vec{r}|=\sqrt{(|\vec{a}|\cos\theta_{a} + |\vec{b}|\cos\theta_{b})^2 + (|\vec{a}|\sin\theta_{a} + |\vec{b}|\cos\theta_{b})^2} \\ +\theta_{r}=\tan^{-1}(\frac{|\vec{a}|\sin\theta_{a} + |\vec{b}|\sin\theta_{b}}{|\vec{a}|\cos\theta_{a} + |\vec{b}|\cos\theta_{b}}) $$ ## Resources From d89b2cc3c692535ba6298c302f8b15ab910e37ae Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 22 Sep 2020 10:46:29 -0400 Subject: [PATCH 5/5] phys: remove decomposition and rephrase for clarity --- docs/sph3u7.md | 60 +++++++++++++++++++++++++++----------------------- 1 file changed, 32 insertions(+), 28 deletions(-) diff --git a/docs/sph3u7.md b/docs/sph3u7.md index 2ca6465..06e53ea 100644 --- a/docs/sph3u7.md +++ b/docs/sph3u7.md @@ -179,9 +179,37 @@ When subtracting a vector, **negate** the vector being subtracted by giving it a (Source: Kognity) +### 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. +$$\vec{F}_x + \vec{F}_y = \vec{F}$$ + +!!! info "Reminder" + The **component form** of a vector is expressed as $(|\vec{a}_x|, |\vec{a}_y|)$ + +(Source: Kognity) + +By using the primary trignometric identities: +$$ +|\vec{a}_{x}| = |\vec{a}|\cos\theta_{a} \\ +|\vec{a}_{y}| = |\vec{a}|\sin\theta_{a} +$$ + +(Source: Kognity) + +Using their component forms, to: + + - add two vectors, add their x- and y-coordinates together. + - subtract two vectors, subtract their x- and y-coordinates together. + +$$ +(a_{x}, a_{y}) + (b_{x}, b_{y}) = (a_{x} + b_{x}, a_{y} + b_{y}) \\ +(a_{x}, a_{y}) - (b_{x}, b_{y}) = (a_{x} - b_{x}, a_{y} - b_{y}) +$$ + ### Parallelogram rule -The parallelogram rule states that the sum of two vectors that form two sides of a parallelogram is the diagonal of that parallelogram. +The parallelogram rule states that the sum of two vectors that form two sides of a parallelogram is the diagonal of that parallelogram. The **sine** and **cosine laws** can be used to solve for the resultant vector. (Source: Kognity) @@ -189,34 +217,10 @@ The parallelogram rule states that the sum of two vectors that form two sides of The product of a vector multiplied by a scalar is a vector with a magnitude of the vector multiplied by the scalar with the same direction as the original vector. -$$v[direction] × s = (v×s)[direction]$$ +$$\vec{v} × s = (|\vec{v}|×s)[\theta_{v}]$$ -### Vector decomposition - -By breaking up a vector into lengths along the x- and y-axes, the sum of two vectors can be calculated algebraically. - -(Source: Kognity) - -For vector $\vec{a}$ and vector $\vec{b}$: - -$$ -\vec{a}_{x} = a\cos\theta_{a} \\ -\vec{a}_{y} = a\sin\theta_{a} -$$ - -Proof: -$|\vec{a}| = \sqrt{a^{2}_{x}+a^{2}_{y}} \\$ -$= \sqrt{(|\vec{a}|\cos\theta_{a})^2 + (|\vec{a}|\sin\theta{a})^2} \\$ -$=\sqrt{|\vec{a}|^2(\cos\theta_{a}^2 + \sin\theta_{a}^2)} \\$ -$=\sqrt{|\vec{a}|^2} \\$ -$=|\vec{a}|$ - -From the diagram above, we can figure out that: - -$$ -|\vec{r}|=\sqrt{(|\vec{a}|\cos\theta_{a} + |\vec{b}|\cos\theta_{b})^2 + (|\vec{a}|\sin\theta_{a} + |\vec{b}|\cos\theta_{b})^2} \\ -\theta_{r}=\tan^{-1}(\frac{|\vec{a}|\sin\theta_{a} + |\vec{b}|\sin\theta_{b}}{|\vec{a}|\cos\theta_{a} + |\vec{b}|\cos\theta_{b}}) -$$ +!!! example + $$3 \text{ m} · 47 \text{ ms}^{-1}[N20°E] = 141 \text{ ms}^{-1}[N20°E]$$ ## Resources