2.3 Vectors

Vectors are groups of numbers which, when named “vectors”, have both a magnitude and direction. The magnitude of the vector corresponds to the length of the vector. The direction of the vector must be relative to a given coordinate system. Within mathematics, vectors are described using a linearly independent basis. A basis is a set of $n$ vectors within a coordinate system. All other vectors can be build using combinations of these $n$ vectors. The term linearly independent signifies that none of the basis vectors are made up of other basis vectors. The general term used to describe these possible combinations of vectors using a basis is called vector space. It is possible to have a vector space build up of any positive amount of basis vectors. However, since we exist in 3 dimensions, in engineering applications we typically operate in a vector space composed of 3 basis vectors. Mathematically, we could say we operate in a vector space of 3. These 3 basis vectors are named $î$, $ĵ$, and $\widehat{k}$ and are shown in equations 2.190, 2.191, and 2.192 respectively.

A vector $\overrightarrow{a}$ operating in the the vector space build on the $î$, $ĵ$, and $\widehat{k}$ vectors would take on the following format shown in equation 2.193.

As mentioned previously, the magnitude of a vector is the length of the vector. Using geometry, we can calculate the magnitude of vector $\overrightarrow{a}$ shown in equation 2.193 using the following equation (the symbol for magnitude is $||a||$).

A unit vector is a vector of magnitude 1. Any vector can be converted to a unit vector by dividing all its components ($a_1$, $a_2$, $a_3$) by its magnitude ($||a||$).

In this book, we will cover two vector operations: the dot product and the cross product.

2.3.1 Vector addition and subtraction

To add two vectors, we simply add each component as shown in equation 2.194.

Similarly, to subtract vectors, we simply subtract each component as shown in equation 2.195.

2.3.2 The Dot Product

The dot product is a vector operation that can be performed on two vectors which exist in any $n$ value vector space. In vector space 3, the dot product is defined as equation 2.196 for vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

Equation 2.196 can be extrapolated to the perform the dot product on vectors which exist in a different vector space.

An alternative equation for calculating the dot product is shown in equation 2.197.

In equation 2.197, $𝜃$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

Orthogonal Vectors are defined as vectors with a dot product of 0.

2.3.3 The Cross Product

The cross product is a vector operation that can be performed on two vectors which exist in a vector space of 3 or more. However, in this book we will focus only on the cross product in vector space 3. In this situation, the cross product is defined as equation 2.198 for $\overrightarrow{a}$ and $\overrightarrow{b}$.

$\overrightarrow{n}$ is defined as a unit vector which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$. $𝜃$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

We can use equation 2.198 to state the following for our vector space 3 basis vectors (equations equations 2.190, 2.191, and 2.192) introduced earlier.

Please note that in equations 2.199, 2.200 and 2.201, a angle convention was introduced (for example, the $𝜃$ between $î$ and $ĵ$ is defined to be positive - this implies that the $𝜃$ between $ĵ$ and $î$ is negative).

Our goal now is to find a general expression for the cross product of vector space 3 vectors $a$ and $b$. This problem statement is presented mathematically as follows.

Distributing, we get the following equation.

We know that any unit vector crossed with itself is 0, therefore, $î\times î$, $ĵ\times ĵ$, $\widehat{k}\times \widehat{k}$ terms are eliminated. Simplifying and replacing the cross products with identities provided in equations 2.199, 2.200, and 2.201, we can simplify to get the following.