1.1. Vector Algebra

For the sake of convenience, Vector is defined as a quantity with both magnitude and direction.

In contrast, a quantity with a magnitude but no direction is defined as Scalar.

 

Mathematically, the vector is an element defined inside a vector space. Details would be discussed in linear algebra, so it is rather convenient to think of a set of every existing vector on 3-dimensional space.  This set has two major algebraic properties.

 

1. Addition is defined.

Vector addition is commutative, associative and there is an addition identity for vectors.

$$\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$$

$$(\mathbf{A}+\mathbf{B})+\mathbf{C}=\mathbf{A}+(\mathbf{B}+\mathbf{C})$$

$$\mathbf{A}+\mathbf{0}=\mathbf{A}$$

Vector subtraction is defined by the addition of inverse vectors. Definition for an inverse vector is discussed later.

$$\mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})$$

 

2. Scalar multiplication is defined.

When multiplied by a scalar, the vector has its magnitude changed but is parallel to the original vector.

Scalar multiplication is associative and there is a scalar multiplication identity for vectors.

$$a(\mathbf{A}+\mathbf{B})=a\mathbf{A}+a\mathbf{B}$$

$$1\mathbf{A}=\mathbf{A}$$

When the scalar is smaller than 0, the direction of the vector flips.

The resulting vector is called an inverse vector of the original vector when the scalar is -1.

 

Multiplication between vectors is not as simply defined as a multiplication between scalars. There are two major operations often used in electromagnetism: dot product and cross product. These two are intuitively more complicated, but it is essential to understand these for future calculations in this study.

 

1. Dot product

Dot product of two vectors is defined as:

$$\mathbf{A}\cdot \mathbf{B}=\left \|\mathbf{A}  \right \|\left \|\mathbf{B}  \right \|\cos \theta $$

θ is the inner angle of the two vectors. Notice that the resulting value is a multiplication of scalars, and therefore a scalar. The dot product is often called scalar product (it is different from scalar multiplication).

 

Dot product is communitative, associative, and distributive.

$$\mathbf{A}\cdot \mathbf{B}=\mathbf{B}\cdot \mathbf{A}$$

$$(\mathbf{A}\cdot \mathbf{B})\cdot \mathbf{C}=\mathbf{A}\cdot(\mathbf{B}\cdot \mathbf{C})$$

$$(\mathbf{A}+ \mathbf{B})\cdot \mathbf{C}=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C}$$

Since cos0=1,

$$\mathbf{A}\cdot \mathbf{A}=\left \| \mathbf{A} \right \|^{2}$$

 

2. Cross product

Cross product of two vectors is defined as:

$$\mathbf{A}\times \mathbf{B}=\left \|\mathbf{A}  \right \|\left \|\mathbf{B}  \right \|\sin\theta \hat{\mathbf{n}}$$

n is a unit vector perpendicular for both vectors. Notice that the resulting value is a vector. The cross product is often called vector product.

 

Cross product is distributive but not associative.

$$(\mathbf{A}+ \mathbf{B})\times \mathbf{C}=\mathbf{A}\times\mathbf{C}+\mathbf{B}\times\mathbf{C}$$

Cross product is anticommunitative.

$$\mathbf{A}\times \mathbf{B}=-\mathbf{B}\times \mathbf{A}$$

Since sin0=0,

$$\mathbf{A}\times \mathbf{A}=\mathbf{0}$$

 

When you actually perform the operation, it is convenient to use the components of the vector inappropriate coordinate system. A vector is defined in a 3-dimensional cartesian coordinate as:

$$\mathbf{A}=A_{x}\mathbf{\hat{x}}+A_{y}\mathbf{\hat{y}}+A_{z}\mathbf{\hat{z}}$$

2.3.1.1. Unit vector notation

Now we can define the four basic operations with the components of vectors.

$$\mathbf{A}+\mathbf{B}=(A_{x}+B_{x})\mathbf{\hat{x}}+(A_{y}+B_{y})\mathbf{\hat{y}}+(A_{z}+B_{z})\mathbf{\hat{z}}$$

2.3.1.2. Vector addition

$$a\mathbf{A}=(aA_{x})\mathbf{\hat{x}}+(aA_{y})\mathbf{\hat{y}}+(aA_{z})\mathbf{\hat{z}}$$

2.3.1.3. Scalar multiplication

$$\mathbf{A}\cdot\mathbf{B}=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}$$

2.3.1.4. Dot product

$$\mathbf{A}\times\mathbf{B}=\begin{vmatrix}
\mathbf{\hat{x}} & \mathbf{\hat{y}} & \mathbf{\hat{z}}\\ 
A_{x} & A_{y} & A_{z}\\ 
B_{x} & B_{y} & B_{z}
\end{vmatrix}$$

2.3.1.5. Cross product

 

These are the properties of the algebraic structure of vectors. Every vector operation is defined based on these algebraic operations, so it is essential to understand these.