vector projections & signed lengths, changed \vec to \boldsymbol

diff --git a/spec/vectors.md b/spec/vectors.md
index f832af7f4eb6f0f3009dc2487daaf2e0e6ff190d..a2122b07af967d273b84716b4a25025dcb83a087 100644 (file)
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -3,8 +3,9 @@ header-includes:
   - \documentclass{standalone}
   - \usepackage{cleveref}
   - \usepackage{harpoon}
   - \documentclass{standalone}
   - \usepackage{cleveref}
   - \usepackage{harpoon}

-  - \usepackage{accent} \newcommand{\vect}[1]{\accentset{\rightharpoonup}{#1}}
----
+  - \usepackage{accent}
+  - \usepackage{amsmath}
+...

 
 # Vectors
 
 
 # Vectors
 


@@ -22,36 +23,36 @@ header-includes:
 
 ## Vector addition
 
 
 ## Vector addition
 

-$\vec{u} + \vec{v}$ can be represented by drawing each vector head to tail then joining the lines.  
+$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines.  

 Addition is commutative (parallelogram)
 
 ## Scalar multiplication
 
 Addition is commutative (parallelogram)
 
 ## Scalar multiplication
 

-For $k \in \mathbb{R}^+$, $k\vec{u}$ has the same direction as $\vec{u}$ but length is multiplied by a factor of $k$.
+For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ but length is multiplied by a factor of $k$.

 
 When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.
 
 ## Vector subtraction
 
 
 When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.
 
 ## Vector subtraction
 

-To find $\vec{u} - \vec{v}$, add $\vec{-v}$ to $\vec{u}$
+To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$

 
 ## Parallel vectors
 
 Parallel vectors have same direction or opposite direction.
 
 
 ## Parallel vectors
 
 Parallel vectors have same direction or opposite direction.
 

-**Two non-zero vectors $\vec{u}$ and $\vec{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\vec{u} = k \vec{v}$**
+**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$**

 
 ## Position vectors
 
 Vectors may describe a position relative to $O$.
 
 
 ## Position vectors
 
 Vectors may describe a position relative to $O$.
 

-For a point $A$, the position vector is $\vec{OA}$
+For a point $A$, the position vector is $\boldsymbol{OA}$

 
 ## Linear combinations of non-parallel vectors
 
 
 ## Linear combinations of non-parallel vectors
 

-If two non-zero vectors $\vec{a}$ and $\vec{b}$ are not parallel, then:
+If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:

 
 

-$$m\vec{a} + n\vec{b} = p \vec{a} + q \vec{b}\quad\text{implies}\quad m = p, \> n = q$$
+$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$

 
 ## Column vector notation
 
 
 ## Column vector notation
 


@@ -59,45 +60,53 @@ A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begi
 
 ## Component notation
 
 
 ## Component notation
 

-A vector $\vec{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\vec{u} = x\vec{i} + y\vec{j}$.  
-$\vec{u}$ is the sum of two components $x\vec{i}$ and $y\vec{j}$  
-Magnitude of vector $\vec{u} = x\vec{i} + y\vec{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$
+A vector $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$.  
+$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$  
+Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$

 
 Basic algebra applies:  
 
 Basic algebra applies:  

-$(x\vec{i} + y\vec{j}) + (m\vec{i} + n\vec{j}) = (x + m)\vec{i} + (y+n)\vec{j}$  
+$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$  

 Two vectors equal if and only if their components are equal.
 
 ## Unit vectors
 
 Two vectors equal if and only if their components are equal.
 
 ## Unit vectors
 

-A vector of length 1. $\vec{i}$ and $\vec{j}$ are unit vectors.
+A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.
+
+A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:

 
 

-A unit vector in direction of $\vec{a}$ is denoted by $\hat{\vec{a}}$
+$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$

 
 

-Also, unit vector of $\vec{a}$ can be defined by $\vec{a} \cdot {|\vec{a}|}$
+Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$

 
 ## Scalar products / dot products
 
 
 ## Scalar products / dot products
 

-If $\vec{a} = a_i \vec{i} + a_2 \vec{j}$ and $\vec{b} = b_i \vec{i} + b_2 \vec{j}$, the dot product is:
-$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$$
+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.
 
 
 Produces a real number, not a vector.
 

-$$\vec{a} \cdot \vec{a} = |\vec{a}|^2$$
+$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$

 
 ## Geometric scalar products
 
 
 ## Geometric scalar products
 

-$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$
+$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$

 
 where $0 \le \theta \le \pi$
 
 ## Perpendicular vectors
 
 
 where $0 \le \theta \le \pi$
 
 ## Perpendicular vectors
 

-If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$ (since $\cos 90 = 0$)
+If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)

 
 ## Finding angle between vectors
 
 
 ## Finding angle between vectors
 

-$$\cos \theta = {{\vec{a} \cdot \vec{b}} \over {|\vec{a}| |\vec{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\vec{a}| |\vec{b}|}}$$
+$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$

 
 
 ## Vector projections
 
 
 
 ## Vector projections
 

+Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$.
+
+$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
+
+
+