Vector Mathematics in 3D

Vector Mathematics in 3D
An Introduction to 3D Vectors

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Some Notation

The Set of Real Numbers:
- \(\mathbb{R}^{1} \doteq \mathbb{R}\)
The Set of Ordered Triples of Real Numbers:
- \(\mathbb{R}^{3} \doteq \mathbb{R} \times \mathbb{R} \times \mathbb{R}\)

Members of \(\mathbb{R}^{3}\)

The Notation You Are Most Familiar With:
- \((x, y, z)\)
A More Convenient Notation:
- \(\bs{p} = (p_{x}, p_{y}, p_{z})\)
A Still More Convenient Notation:
- \(\bs{p} = (p_{1}, p_{2}, p_{3})\)

Two Things to be Aware Of

Orientation:
- It sometimes matters whether we think of the ordered pair as a "row" or a "column", though we won't distinguish for now
- An example of a "row": \(\bs{p} = [p_1 \quad p_2 \quad p_3]\)
- An example of a "column": \(\bs{p} = \left[ \begin{array}{c}p_1 \\ p_2 \\ p_3 \end{array} \right]\)
Points and Directions:
- In 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

Visualization

Points

Visualization (cont.)

Directions

Vector Fields

Definition:
- A vector field is a map \(F: A \subset \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) that assigns to each point \(\bs{p} \in A\) a vector \(F(\bs{p})\)
Visualization:

Multiplication by a Scalar

Definition:
- Given \(\alpha \in \mathbb{R}\) and \(\bs{p} \in \mathbb{R}^{3}\)
- \(\alpha \bs{p} = (\alpha p_{1}, \alpha p_{2}, \alpha p_{3})\)
Visualization:

Addition

Definition:
- Given \(\bs{q} \in \mathbb{R}^{3}\) and \(\bs{r} \in \mathbb{R}^{3}\)
- \(\bs{q} + \bs{r} = (q_{1}+r_{1}, q_{2}+r_{2}, q_{3}+r_{3})\)
Visualization:

Subtraction

Definition:
- Given \(\bs{q} \in \mathbb{R}^{3}\) and \(\bs{r} \in \mathbb{R}^{3}\)
- \(\bs{q} - \bs{r} = (q_{1}-r_{1}, q_{2}-r_{2}, q_{3}-r_{3})\)
Visualization:

The Standard Basis

Three Important Vectors:
- \(\bs{i} = (1, 0, 0)\)
- \(\bs{j} = (0, 1, 0)\)
- \(\bs{k} = (0, 0, 1)\)
A General Representation of Vectors in \(\mathbb{R}^3\):
- Suppose \(\bs{q} = (q_{1}, q_{2}, q_{3})\), then
- \(\bs{q} = q_{1} \bs{i} + q_{2} \bs{j} + q_{3} \bs{k}\)

The Standard Basis (cont.)

Visualization

The Standard Basis (cont.)

An Important Observation:
- Every vector \(\bs{p}\) can be written as \(\bs{p} = \alpha \bs{i} + \beta \bs{j} + \gamma \bs{k}\) when \(\alpha\), \(\beta\) and \(\gamma\) are chosen appropriately
As in Two Dimensions:
- \(\bs{p}\) is then said to be a linear combination of \(\bs{i}\), \(\bs{j}\), and \(\bs{k}\)
- \(\bs{i}\), \(\bs{j}\), and \(\bs{k}\) are said to be linearly independent

The Inner (Dot) Product

Two Notations:
- \(\bs{p} \cdot \bs{q}\)
- \(\lt \bs{p}, \bs{q} \gt\)
Definition:
- Given \(\bs{p} \in \mathbb{R}^{3}\) and \(\bs{q} \in \mathbb{R}^{3}\)
- \(\bs{p} \cdot \bs{q} = \sum_{i=1}^{3} p_{i} q_{i} = p_{1} q_{1} + p_{2} q_{2} + p_{3} q_{3} \)

The Euclidean Norm (Length) of a Vector

The (Euclidean) Length of a Vector:
- \(||\bs{p}|| = \sqrt{\sum_{i=1}^{3} p_{i}^{2}} = \sqrt{p_{1}^{2} + p_{2}^2 + p_{2}^2}\)
An Observation:
- \(\bs{p} \cdot \bs{p} = p_{1}^{2} + p_{2}^2 + p_{3}^2\)
Substituting:
- \(||\bs{p}|| = (\bs{p} \cdot \bs{p})^{\frac{1}{2}}\)

Unit Vector

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

Normalization

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:
- \( \left\| \frac{\bs{p}}{||\bs{p}||} \right\| = 1 \)

The Angle Formed by Vectors

Getting Started:
The Plane Formed by the Vectors:
- \(\bs{p}\) and \(\bs{q}\) define a plane
- So, what we learned about 2D vectors applies here
From the Law of Cosines:
- \(\theta = \cos^{-1} \frac{\bs{p} \cdot \bs{q}}{||\bs{p}|| \enspace||\bs{q}||}\)

Cross Product

Definition:
- Given two vectors \(\bs{v}, \bs{w} \in \mathbb{R}^{3}\), the vector \(\bs{u} \in \mathbb{R}^{3}\) defined as:
- \(\bs{u} = (v_{2}w_{3} - w_{2}v_{3}, v_{3}w_{1} - w_{3}v_{1}, v_{1}w_{2} - w_{1}v_{2})\)
- is called the cross product of \(\bs{v}\) and \(\bs{w}\)
An Example:
- Let \(\bs{v} = \left[ \begin{array}{c}1 \\ 0 \\ 2\end{array}\right]\) and \(\bs{w} = \left[ \begin{array}{c}5 \\ 3 \\ 4\end{array}\right]\). Then:
- \(\bs{u} = \bs{v} \times \bs{w} = \left[ \begin{array}{c}0 \cdot 4 - 3 \cdot 2 \\ 2 \cdot 5 - 4 \cdot 1 \\ 1 \cdot 3 - 5 \cdot 0\end{array}\right] = \left[ \begin{array}{c}-6 \\ 6 \\ 3\end{array}\right]\)
Be Careful:
- Unlike the inner (or dot) product, the cross product is a vector
- Don't confuse this with the cross product of two sets
- Don't confuse the different uses of the symbol \(\times\)

Cross Product (cont.)

A Simple Case:
- Suppose both vectors are in the \(xy\)-plane so that:
- \(\bs{v} = a \bs{i} + b \bs{j} = \left[ \begin{array}{c}a \\ b \\ 0\end{array}\right]\) and \(\bs{w} = d \bs{i} + e \bs{j} = \left[ \begin{array}{c}d \\ e \\ 0\end{array}\right]\)
- Hence:
- \(u = \bs{v} \times \bs{w} = \left[ \begin{array}{c}0 \\ 0 \\ ae - bd\end{array}\right]\)
Visualization:
- \(a = ||\bs{v}|| \cos \gamma\)
- \(b = ||\bs{v}|| \sin \gamma\)
- \(d = ||\bs{w}|| \cos \beta\)
- \(e = ||\bs{w}|| \sin \beta\)
A Simple Derivation:
- (1) \(|| \bs{v} \times \bs{w} || = | ae - bd |\)
- (2) \(|| \bs{v} \times \bs{w} || = | \enspace ||\bs{v}|| \cos \gamma \cdot ||\bs{w}|| \sin \beta - ||\bs{v}|| \sin \gamma \cdot ||\bs{w}|| \cos \beta \enspace | \)
- Since \(\sin (\beta - \gamma) = (\sin \beta \cos \gamma - \cos \beta \sin \gamma)\) it follows that:
- (3) \(|| \bs{v} \times \bs{w} || = ||\bs{v}|| \enspace ||\bs{w}|| \sin(\beta - \gamma) = ||\bs{v}|| \enspace ||\bs{w}|| \sin \theta\)

Cross Product (cont.)

The Visualization Revisited:
An Interpretation:
- \(|| \bs{v} \times \bs{w} || = ||\bs{v}|| \enspace ||\bs{w}|| \sin \theta\) which is the (unsigned) area of the parallelogram formed by \(\bs{v}\), \(\bs{w}\), and \((\bs{v}+\bs{w})\)

Cross Products (cont.)

An Important Property:
- \(\bs{v} \times \bs{w}\) is orthogonal to both \(\bs{v}\) and \(\bs{w}\) (and, hence, all vectors in the plane they form)
Proof:
- \((\bs{v} \times \bs{w}) \cdot \bs{v} = (v_1 v_2 w_3 - v_1 w_2 v_3) + (w_1 v_2 v_3 - v_1 v_2 w_3) + (v_1 w_2 v_3 - w_1 v_2 v_3) = 0\)
- \((\bs{v} \times \bs{w}) \cdot \bs{w} = (w_1 v_2 w_3 - w_1 w_2 v_3) + (w_1 w_2 v_3 - v_1 w_2 w_3) + (v_1 w_2 w_3 - w_1 v_2 w_3) = 0\)
An Observation:
- This property is sometimes used as the defition of the cross product

Normal Vectors

Definition:
- A vector that is perpendicular to a surface
An Observation:
- In three dimensions a normal can point either "inward" or "outward"
The Right Hand Rule for Cross Products:
- Point the fingers of your right hand in the direction of the first vector and curl them around to the second vector. Your thumb will point in the positive direction.
An Alternative Definition of the Cross Product:
- \(\bs{v} \times \bs{w} = ||\bs{v}|| \cdot ||\bs{w}|| \cdot \sin(\theta) \cdot \bs{n}\) where \(\bs{n}\) is the unit normal to the plane

Scalar Triple Product

Definition:
- Given \(\bs{v}, \bs{w}, \bs{u} \in \mathbb{R}^{3}\), the scalar triple product is defined as:
- \(\bs{u} \cdot (\bs{v} \times \bs{w})\)
Visualization:
- The area of the parallelogram formed by \(v\) and \(\bs{w}\) is \(||\bs{v} \times \bs{w}||\)
- The volume of the parallelepiped is \(||\bs{u}|| \cos \alpha ||\bs{v} \times \bs{w}||\)
- From the Law of Cosines: \(||\bs{a}|| \cos \alpha ||\bs{b}|| = \bs{a} \cdot \bs{b}\)
- So, the volume of the parallelepiped is \(\bs{u} \cdot (\bs{v} \times \bs{w})\)

The Cross Product Revisited

A Special Case:
- Replace \(\bs{u}\) with the unit normal \(n\)
Visualization:
Interpretation:
- Given the definition of \(\bs{n}\) we know that \(||\bs{n}|| = 1\)
- So, ignoring units, the volume of this parallelepiped is the same as the area of the parallelogram
- We can use \(||\bs{n}|| \cos \alpha ||\bs{v} \times \bs{w}|| = \bs{n} \cdot (\bs{v} \times \bs{w})\) as the signed area of the parallelogram

There's Always More to Learn