# Vectors/Vector Algebra

## Vector Algebra Operations

If $\mathbf {a} \,$  and $\mathbf {b} \,$  are vectors, then the sum $\mathbf {c} =\mathbf {a} +\mathbf {b} \,$  is also a vector.

The two vectors can also be subtracted from one another to give another vector $\mathbf {d} =\mathbf {a} -\mathbf {b} \,$ .

### Multiplication by a scalar

Multiplication of a vector $\mathbf {b} \,$  by a scalar $\lambda \,$  has the effect of stretching or shrinking the vector.

You can form a unit vector ${\hat {\mathbf {b} }}\,$  that is parallel to $\mathbf {b} \,$  by dividing by the length of the vector $|\mathbf {b} |\,$ . Thus,

${\hat {\mathbf {b} }}={\frac {\mathbf {b} }{|\mathbf {b} |}}~.$

### Scalar product of two vectors

The scalar product or inner product or dot product of two vectors is defined as

$\mathbf {a} \cdot \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\cos(\theta )$

where $\theta \,$  is the angle between the two vectors (see Figure 2(b)).

If $\mathbf {a} \,$  and $\mathbf {b} \,$  are perpendicular to each other, $\theta =\pi /2\,$  and $\cos(\theta )=0\,$ . Therefore, ${\mathbf {a} }\cdot {\mathbf {b} }=0$ .

The dot product therefore has the geometric interpretation as the length of the projection of $\mathbf {a} \,$  onto the unit vector ${\hat {\mathbf {b} }}\,$  when the two vectors are placed so that they start from the same point (tail-to-tail).

The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as

${\mathbf {a} }\cdot {\mathbf {b} }=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=\sum _{i=1..3}a_{i}b_{i}~.$

If the vector is $n$  dimensional, the dot product is written as

${\mathbf {a} }\cdot {\mathbf {b} }=\sum _{i=1..n}a_{i}b_{i}~.$

Using the Einstein summation convention, we can also write the scalar product as

${\mathbf {a} }\cdot {\mathbf {b} }=a_{i}b_{i}~.$

Also notice that the following also hold for the scalar product

1. ${\mathbf {a} }\cdot {\mathbf {b} }={\mathbf {b} }\cdot {\mathbf {a} }$  (commutative law).
2. ${\mathbf {a} }\cdot {(\mathbf {b} +\mathbf {c} )}={\mathbf {a} }\cdot {\mathbf {b} }+{\mathbf {a} }\cdot {\mathbf {c} }$  (distributive law).

### Vector product of two vectors

The vector product (or cross product) of two vectors $\mathbf {a} \,$  and $\mathbf {b} \,$  is another vector $\mathbf {c} \,$  defined as

$\mathbf {c} ={\mathbf {a} }\times {\mathbf {b} }=|\mathbf {a} ||\mathbf {b} |\sin(\theta ){\hat {\mathbf {c} }}$

where $\theta \,$  is the angle between $\mathbf {a} \,$  and $\mathbf {b} \,$ , and ${\hat {\mathbf {c} }}\,$  is a unit vector perpendicular to the plane containing $\mathbf {a} \,$  and $\mathbf {b} \,$  in the right-handed sense.

In terms of the orthonormal basis $(\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})\,$ , the cross product can be written in the form of a determinant

${\mathbf {a} }\times {\mathbf {b} }={\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}}.$

In index notation, the cross product can be written as

${\mathbf {a} }\times {\mathbf {b} }\equiv e_{ijk}a_{j}b_{k}.$

where $e_{ijk}$  is the Levi-Civita symbol (also called the permutation symbol, alternating tensor).

## Identities from Vector Algebra

Some useful vector identities are given below.

1. ${\mathbf {a} }\times {\mathbf {b} }=-{\mathbf {b} }\times {\mathbf {a} }$ .
2. ${\mathbf {a} }\times {\mathbf {b} +\mathbf {c} }={\mathbf {a} }\times {\mathbf {b} }+{\mathbf {a} }\times {\mathbf {c} }$ .
3. ${\mathbf {a} }\times {({\mathbf {b} }\times {\mathbf {c} })}=\mathbf {b} ({\mathbf {a} }\cdot {\mathbf {c} })-\mathbf {c} ({\mathbf {a} }\cdot {\mathbf {b} })$
4. ${({\mathbf {a} }\times {\mathbf {b} })}\times {\mathbf {c} }=\mathbf {b} ({\mathbf {a} }\cdot {\mathbf {c} })-\mathbf {a} ({\mathbf {b} }\cdot {\mathbf {c} })$
5. ${\mathbf {a} }\times {\mathbf {a} }=\mathbf {0}$
6. ${\mathbf {a} }\cdot {({\mathbf {a} }\times {\mathbf {b} })}={\mathbf {b} }\cdot {({\mathbf {a} }\times {\mathbf {b} })}=\mathbf {0}$
7. ${({\mathbf {a} }\times {\mathbf {b} })}\cdot {\mathbf {c} }={\mathbf {a} }\cdot {({\mathbf {b} }\times {\mathbf {c} })}$

For more details on the topics of this chapter, see Vectors in the wikibook on Calculus.