• 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