# Vectors

A vector is an *object* that has certain properties. What are these properties? We usually say that these properties are:

- a vector has a magnitude (or length)
- a vector has a direction.

To make the definition of the vector object more precise we may also say that vectors are objects that satisfy the properties of a vector space.

The standard notation vector can be represented in component form with respect to the *basis* (e1,e2) as

- a=a1e1+a2e2

where e1 and e2 are *orthonormal unit vectors*. Orthonormal means they are at right angles to each other (*orthogonal*) and are unit vectors. Recall that *unit vectors* are vectors of length 1. These vectors are also called *basis* vectors.

You could also represent the same vector a in terms of another set of basis vectors (g1,g2) as shown in Figure 1(b). In that case, the components of the vector are (b1,b2) and we can write

- a=b1g1+b2g2 .

Note that the basis vectors g1 and g2 do not necessarily have to be unit vectors. All we need is that they be *linearly independent*, that is, it should not be possible for us to represent one solely in terms of the others.

In three dimensions, using an *orthonormal basis*, we can write the vector a as

- a=a1e1+a2e2+a2e3

where e3 is perpendicular to both e1 and e2. This is the usual basis in which we express arbitrary vectors.

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## Vector Algebra Operations

### Addition and Subtraction

If a and b are vectors, then the sum c=a+b is also a vector (see Figure 2(a)).

The two vectors can also be subtracted from one another to give another vector d=a−b.

### Multiplication by a scalar

Multiplication of a vector b by a scalar λ has the effect of stretching or shrinking the vector (see Figure 2(b)).

You can form a unit vector Missing argument for mathbf that is parallel to b by dividing by the length of the vector |b|. Thus,

- Missing argument for mathbf

### Scalar product of two vectors

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

- a⋅b=|a||b|cos(θ)

where θ is the angle between the two vectors (see Figure 2(b)).

If a and b are perpendicular to each other, θ=π/2 and cos(θ)=0. Therefore, a⋅b=0.

The dot product therefore has the geometric interpretation as the length of the projection of a onto the unit vector Missing argument for mathbf when the two vectors are placed so that they start from the same point.

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

- a⋅b=a1b1+a2b2+a3b3=∑i=1..3aibi .

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

- a⋅b=∑i=1..naibi .

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

- a⋅b=aibi .

Also notice that the following also hold for the scalar product

- a⋅b=b⋅a (commutative law).
- a⋅(b+c)=a⋅b+a⋅c (distributive law).

### Vector product of two vectors

The *vector* product (or *cross* product) of two vectors a and b is another vector c defined as

- c=a×b=|a||b|sin(θ)c^

where θ is the angle between a and b, and c^ is a unit vector perpendicular to the plane containing a and b in the right-handed sense (see Figure 3 for a geometric interpretation)

In terms of the orthonormal basis (e1,e2,e3), the cross product can be written in the form of a determinant

- a×b=∣∣∣∣e1a1b1e2a2b2e3a3b3∣∣∣∣ .

In index notation, the cross product can be written as

- a×b≡eijkajbk .

where eijk is the Levi-Civita symbol (also called the permutation symbol, alternating tensor).

## Identities from Vector Algebra

Some useful vector identities are given below.

- a×b=−b×a.
- a×b+c=a×b+a×c.
- a×(b×c)=b(a⋅c)−c(a⋅b) ~.
- (a×b)×c=b(a⋅c)−a(b⋅c) ~.
- a×a=0~.
- a⋅(a×b)=b⋅(a×b)=0~.
- (a×b)⋅c=a⋅(b×c)~.