From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 6 Mar 2019 11:00:39 +0000 (+1100)
Subject: [spec] finish vectors reference
X-Git-Tag: yr12~217
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/aa43df379be8af5784531ada2685585ffdb33f02

[spec] finish vectors reference
---

diff --git a/spec/graphics/vectors-3d.png b/spec/graphics/vectors-3d.png
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diff --git a/spec/vectors-ref.pdf b/spec/vectors-ref.pdf
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diff --git a/spec/vectors.md b/spec/vectors.md
index 1ac62fe..932f614 100644
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -24,7 +24,7 @@ header-includes:
 - vectors with equal magnitude and direction are equivalent
 
 
-[//]: # ![](graphics/vectors-intro.png){#id .class width=20%} 
+![](graphics/vectors-intro.png){#id .class width=20%} 
 
 ## Vector addition
 
@@ -43,9 +43,9 @@ To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol
 
 ## Parallel vectors
 
-Parallel vectors have same direction or opposite direction.
+Same or opposite direction
 
-**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$**
+$$\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}$$ 
 
 ## Position vectors
 
@@ -78,25 +78,14 @@ Basic algebra applies:
 $(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$  
 Two vectors equal if and only if their components are equal.
 
-## Unit vectors
+## Unit vector $|\hat{\boldsymbol{a}}|=1$
 
-A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.
+\begin{equation}\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\end{equation}
 
-A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:
+## Scalar/dot product $\boldsymbol{a} \cdot \boldsymbol{b}$
 
-$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$
-
-Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$
-
-## Scalar products / dot products
-
-If $\boldsymbol{a} = a_i \boldsymbol{i} + a_2 \boldsymbol{j}$ and $\boldsymbol{b} = b_i \boldsymbol{i} + b_2 \boldsymbol{j}$, the dot product is:
 $$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$
 
-Produces a real number, not a vector.
-
-$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
-
 **on CAS:** `dotP([a b c], [d e f])`
 
 ## Scalar product properties
@@ -132,22 +121,42 @@ $$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |
 
 **on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)
 
+## Angle between vector and axis
+
+Direction of a vector can be given by the angles it makes with $\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}$ directions.
+
+For $\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
+$$\cos \alpha = {a_1 \over |\boldsymbol{a}|}, \quad \cos \beta = {a_2 \over |\boldsymbol{a}|}, \quad \cos \gamma = {a_3 \over |\boldsymbol{a}|}$$
+
+**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}$ and $x$-axis
 
 ## Vector projections
 
-Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$.
+Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$:
 
 $$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 
-Scalar resolute of $\vec{a}$ on $\vec{b} = |\vec{u}| = \vec{a} \cdot \hat{\vec{b}}$ (results in a scalar)  
-Vector resolute of $\vec{a}$ perpendicular to $b$ is equal to $\vec{a} - \vec{u}$ where $\vec{u}$ is vector projection of $\vec{a}$ on $\vec{b}$
+## Scalar resolute of $\boldsymbol{a}$ on $\boldsymbol{b}$
+
+$$r_s = |\boldsymbol{u}| = \boldsymbol{a} \cdot \hat{\boldsymbol{b}}$$
+
+## Vector resolute of $\boldsymbol{a} \perp \boldsymbol{b}$
+
+$$\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u} \> \text{ where } \boldsymbol{u} \text{ is projection } \boldsymbol{a} \text{ on } \boldsymbol{b}$$
 
 ## Vector proofs
 
-**Concurrent lines -** $\ge$ 3 lines intersect at a single point  
-**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 < \mu < 1$)
+### Concurrent lines
+
+$\ge$ 3 lines intersect at a single point  
+
+### Collinear points
+
+$\ge$ 3 points lie on the same line  
+$\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 < \mu < 1$  
+Points $A, B, C$ are collinear iff $\vec{AC}=m\vec{AB} \text{ where } m \ne 0$
 
-Useful vector properties:
+### Useful vector properties
 
 - If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
 - If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
@@ -156,28 +165,23 @@ Useful vector properties:
 
 ## Linear dependence
 
-Vectors $\vec{a}, \vec{b}, \vec{c}$ are linearly dependent if they are non-parallel and:
+Vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ are linearly dependent if they are non-parallel and:
 
-$$k\vec{a}+l\vec{b}+m\vec{c} = 0$$
-$$\therefore \vec{c} = m\vec{a} + n\vec{b} \quad \text{(simultaneous)}$$
+$$k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0$$
+$$\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}$$
 
-$\vec{a}, \vec{b},$ and $\vec{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel.
+$\boldsymbol{a}, \boldsymbol{b},$ and $\boldsymbol{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel.
 
-Vector $\vec{w}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$
+Vector $\boldsymbol{w}$ is a linear combination of vectors $\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}$
 
 ## Three-dimensional vectors
 
-Right-hand rule for axes - $z$ is up or out of page.
-
-## Angle between vector and axis
+Right-hand rule for axes: $z$ is up or out of page.
 
-Direction of a vector can be given by the angles it makes with $\vec{i}, \vec{j}, \vec{k}$ directions.
+i![](graphics/vectors-3d.png)
 
-For $\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
-$$\cos \alpha = {a_1 \over |\vec{a}|}, \quad \cos \beta = {a_2 \over |\vec{a}|}, \quad \cos \gamma = {a_3 \over |\vec{a}|}$$
+## Parametric vectors
 
-**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\vec{i} + b\vec{j} + c\vec{k}$ and $x$-axis
+Parametric equation of line through point $(x_0, y_0, z_0)$ and parallel to $a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}$ is:
 
-## Collinearity
-
-Points $A, B, C$ are collinear iff $\vec{AC}=m\vec{AB} \text{ where } m \ne 0$
+\begin{equation}\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\end{equation}