• The Set of Real Numbers:
  • $\mathbb{R}^{1} \doteq \mathbb{R}$
• The Set of Ordered Pairs of Real Numbers:
  • $\mathbb{R}^{2} \doteq \mathbb{R} \times \mathbb{R}$

• The Notation You Are Most Familiar With:
  • $(x, y)$
• A More Convenient Notation:
  • $\bs{p} = (p_{x}, p_{y})$
• A Still More Convenient Notation:
  • $\bs{p} = (p_{1}, p_{2})$

• Orientation:
  • It is sometimes important to distinguish between "rows" and "columns"
  • An example of a "row": $\bs{p} = [p_1 \quad p_2]$
  • An example of a "column": $\bs{p} = \left[ \begin{array}{c}p_1 \\ p_2 \end{array} \right]$
  • We won't distinguish for now
• Points and Directions:
  • It is sometimes important to distinguish between different kinds of ordered pairs (e.g., points and directions)
  • We won't distinguish for now except in visualizations

• Definition:
  • Given $\alpha \in \mathbb{R}$ and $\bs{p} \in \mathbb{R}^{2}$
  • $\alpha \bs{p} = (\alpha p_{1}, \alpha p_{2})$
• Example:
  • $2 (1,1) = (2,2)$
• Visualization:
  • 

• Definition:
  • Given $\bs{q} \in \mathbb{R}^{2}$ and $\bs{r} \in \mathbb{R}^{2}$
  • $\bs{q} + \bs{r} = (q_{1}+r_{1}, q_{2}+r_{2})$
• Example:
  • $(1.5, -0.5) + (0.5, -1.0) = (2, -1.5)$
• Visualization:
  • 

• Definition:
  • Given $\bs{q} \in \mathbb{R}^{2}$ and $\bs{r} \in \mathbb{R}^{2}$
  • $\bs{q} - \bs{r} = (q_{1}-r_{1}, q_{2}-r_{2})$
• Example:
  • $(1.5, 0.5) - (0.5, 1.0) = (1.0, -0.5)$
• Visualization:
  • 

• Another Visualization:
  • 
• Interpretation:
  • The direction vector points from $\bs{r}$ to $\bs{q}$
  • So, $\bs{r} + (\bs{q} - \bs{r}) = \bs{q}$

• Example:
  • $(1.5, -0.5) + (0.5, -1.0) = (2, -1.5)$
• Visualization:
  • 

• Vector Fields:
  • A vector field is a map $F: A \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ that assigns to each point $p \in A$ a vector $F(p)$
• Visualization of Vector Fields:
  • 

• Two Important Vectors:
  • $\bs{i} = (1, 0)$
  • $\bs{j} = (0, 1)$
• A General Representation of Vectors in $\mathbb{R}^2$
  • Suppose $\bs{q} = (q_{1}, q_{2})$, then
  • $\bs{q} = q_{1} \bs{i} + q_{2} \bs{j}$

• An Important Observation:
  • Every vector $\bs{p}$ can be written as $\bs{p} = \alpha \bs{i} + \beta \bs{j}$ when $\alpha$ and $\beta$ are chosen appropriately
• Some Terminology:
  • $\bs{p}$ is then said to be a linear combination of $\bs{i}$ and $\bs{j}$ (more later)
  • $\bs{i}$ and $\bs{j}$ are said to be linearly independent (more later)

Since the distance from the origin to $\bs{p}$ is 1, it follows that: $$\cos(\theta) = \frac{p_1}{1} \\ \sin(\theta) = \frac{p_2}{1}$$

Which is to say that $\bs{p} := (p_1, p_2) = (\cos(\theta), \sin(\theta))$

Hence, any point, $\bs{q}$, on the unit sphere can be represented as: $$\bs{q} = (1, 0) \cdot \cos(\theta) + (0, 1) \cdot \sin(\theta) = \bs{i} \cos(\theta) + \bs{j} \sin(\theta)$$

• Two Notations:
  • $\bs{p} \cdot \bs{q}$
  • $\lt \bs{p}, \bs{q} \gt$
• Definition:
  • Given $\bs{p} \in \mathbb{R}^{2}$ and $\bs{q} \in \mathbb{R}^{2}$
  • $\bs{p} \cdot \bs{q} = p_{1} q_{1} + p_{2} q_{2}$
• Example:
  • $(2,3) \cdot (4,5) = 2 (4) + 3 (5) = 8 + 15 = 23$

• An Example:
  • 
• Pythagorean Theorem:
  • Length of $\bs{p}$ is $\sqrt{4^{2} + 3^2} = \sqrt{25} = 5$

• The (Euclidean) Length of a Vector:
  • $\sqrt{p_{1}^{2} + p_{2}^2}$
• An Observation:
  • $\bs{p} \cdot \bs{p} = p_{1}^{2} + p_{2}^2$
• Substituting:
  • The length of $\bs{p}$ is $(\bs{p} \cdot \bs{p})^{\frac{1}{2}}$ which we denote by $||\bs{p}||$

• Definition:
  • A vector with a norm (i.e., length) of 1
• Two Examples Revisited:
  • $\bs{i} = (1, 0)$
  • $\bs{j} = (0, 1)$

• Definition:
  • The normalization of a vector $\bs{p}$ is $\frac{\bs{p}}{||\bs{p}||}$ (i.e., the vector divided by its norm)
• An Observation:
  • $||\bs{p}||$ is a scalar so normalization involves multiplication by a scalar
• Relationship to Unit Vectors:
  • $\begin{align*} \left\| \frac{\bs{p}}{||\bs{p}||} \right\| &= \left(\frac{\bs{p}}{||\bs{p}||} \cdot \frac{\bs{p}}{||\bs{p}||}\right)^\frac{1}{2} \\ &= \left[ \left(\frac{1}{||\bs{p}||}p_{1}, \frac{1}{||\bs{p}||}p_{2}\right) \cdot \left(\frac{1}{||\bs{p}||}p_{1}, \frac{1}{||\bs{p}||}p_{2}\right)\right]^\frac{1}{2} \\ &= \left[\frac{1}{||\bs{p}||^{2}}p_{1}^{2} + \frac{1}{||\bs{p}||^{2}}p_{2}^{2}\right]^\frac{1}{2} \\ &= \left[\frac{p_{1}^{2} + p_{2}^{2}}{||\bs{p}||^{2}}\right]^\frac{1}{2} \\ &= \left[\frac{p_{1}^{2} + p_{2}^{2}}{\left(\bs{p} \cdot \bs{p}\right)}\right]^\frac{1}{2} \\ &= \left[\frac{p_{1}^{2} + p_{2}^{2}}{p_{1}^{2} + p_{2}^{2}}\right]^\frac{1}{2} \\ &= 1 \end{align*}$

• Visualization:
  • 
• The Law of Cosines:
  • $||\bs{q} - \bs{r}||^{2} = ||\bs{q}||^{2} + ||\bs{r}||^{2} - 2 ||\bs{q}|| \enspace ||\bs{r}|| \cos \theta$

• From the Law of Cosines:
  • $\cos \theta = \frac{||\bs{q}||^{2} + ||\bs{r}||^{2} - ||\bs{q} - \bs{r}||^{2}}{2 ||\bs{q}|| \enspace ||\bs{r}||} = \frac{\bs{q} \cdot \bs{r}}{||\bs{q}|| \enspace ||\bs{r}||}$
  • $\theta = \cos^{-1} \frac{\bs{q} \cdot \bs{r}}{||\bs{q}|| \enspace ||\bs{r}||}$
• Interpretation:
  • The angle formed by two vectors is related to their inner product
• Special Cases:
  • The two vectors are orthogonal (i.e., the angle is 90 degrees)
  • The two vectors are linearly independent if they are not parallel

• One Perpendicular Vector:
  • 
• Definition:
  • $\bs{q}^{\perp} = (- q_{2}, q_{1})$

• Linear Combination (Revisited):
  • Given two linearly independent vectors $\bs{v}$ and $\bs{w}$ and two scalars $\alpha$ and $\beta$, $\alpha \bs{v} + \beta \bs{w}$ is said to be a linear combination of $\bs{v}$ and $\bs{w}$
• Barycentric Combination:
  • A linear combination in which the weights sum to 1
  • Hence, given two points $\bs{q}$ and $\bs{r}$ and a scalar $\alpha$, $\alpha \bs{q} + (1 - \alpha) \bs{r}$ is said to be a barycentric combination of $\bs{q}$ and $\bs{r}$
• Convex Combination:
  • A linear combination in which the weights sum to 1 and are all less than or equal to 1
  • Hence, given two points $\bs{q}$ and $\bs{r}$ and a scalar $\alpha \in [0, 1]$, $\alpha \bs{q} + (1 - \alpha) \bs{r}$ is said to be a convex combination of $\bs{q}$ and $\bs{r}$

• Barycentric Combination of Three Points:
  • Given three points $\bs{r}$, $\bs{s}$ and $\bs{t}$ and three scalars $\rho$, $\sigma$, and $\tau$ with $\rho + \sigma + \tau = 1$, $\rho \bs{r} + \sigma \bs{s} + \tau \bs{t}$ is said to be a barycentric combination of the three points
• Visualization:
  • 
• An Observation:
  • We can always determine one of the scalars from the other two