From 4393424e4c8c43c73a8ec71f84f1272d267194fe Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 22 Sep 2020 09:58:39 -0400 Subject: [PATCH] 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