tables: yes
author: Andrew Lorimer
classoption: twocolumn
+header-includes:
+- \usepackage{harpoon}
+- \usepackage{amsmath}
+- \pagenumbering{gobble}
---
- **vector:** a directed line segment
- arrow indicates direction
- length indicates magnitude
-- notated as $\vec{a}, \widetilde{A}$
+- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
- column notation: $\begin{bmatrix}
x \\ y
\end{bmatrix}$
- vectors with equal magnitude and direction are equivalent
-
+{#id .class width=20%}
## Vector addition
## 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
Vectors may describe a position relative to $O$.
-For a point $A$, the position vector is $\boldsymbol{OA}$
+For a point $A$, the position vector is $\overrightharp{OA}$
+
+\vfill\eject
## Linear combinations of non-parallel vectors
If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:
-$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$
+$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$
+
+{#id .class width=20%}
+{#id .class width=10%}
## Column vector notation
$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$
Two vectors equal if and only if their components are equal.
-## Unit vectors
-
-A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.
-
-A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:
+## Unit vector $|\hat{\boldsymbol{a}}|=1$
-$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$
+\begin{equation}\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\end{equation}
-Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$
+## Scalar/dot product $\boldsymbol{a} \cdot \boldsymbol{b}$
-## 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
1. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
2. $\boldsymbol{a \cdot 0}=0$
3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
+4. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
+5. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
+6. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$
For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:
-$\boldsymbol{a \cdot b}=\{
- \begin{array}{ll}
- |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction} \\
- -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} \\
- \end{array}$
+$$\boldsymbol{a \cdot b}=\begin{cases}
+|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
+-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
+\end{cases}$$
## Geometric scalar products
## Finding angle between vectors
+**positive direction**
+
$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
+**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 $\boldsymbol{a}$ 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
+### Concurrent lines
+
+$\ge$ 3 lines intersect at a single point
-Useful vector properties:
+### 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
- 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
- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
+## Linear dependence
+
+Vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ are linearly dependent if they are non-parallel and:
+
+$$k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0$$
+$$\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}$$
+
+$\boldsymbol{a}, \boldsymbol{b},$ 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 $\boldsymbol{w}$ 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.
+
+i
+## Parametric vectors
+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:
+\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}