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
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
where e3 is perpendicular to both e1 and e2. This is the usual basis in which we express arbitrary vectors.
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
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
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×c)=b(a⋅c)−c(a⋅b) ~.
- (a×b)×c=b(a⋅c)−a(b⋅c) ~.