Vectors/Vector Algebra

Two vectors are added head to tail.

Vector Algebra OperationsEdit

Addition and SubtractionEdit

If   and   are vectors, then the sum   is also a vector.

The two vectors can also be subtracted from one another to give another vector  .

Multiplication by a scalarEdit

 
Multiplicaton by 2 doubles the length of a vector

Multiplication of a vector   by a scalar   has the effect of stretching or shrinking the vector.

You can form a unit vector   that is parallel to   by dividing by the length of the vector  . Thus,

 

Scalar product of two vectorsEdit

 
The scalar product depends on the cosine of the angle between two vectors.

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

 

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

If   and   are perpendicular to each other,   and  . Therefore,  .

The dot product therefore has the geometric interpretation as the length of the projection of   onto the unit vector   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

 

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

 

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

 

Also notice that the following also hold for the scalar product

  1.   (commutative law).
  2.   (distributive law).

Vector product of two vectorsEdit

 
The area of the parallelogram generated by two vectors is the length of their cross product

The vector product (or cross product) of two vectors   and   is another vector   defined as

 

where   is the angle between   and  , and   is a unit vector perpendicular to the plane containing   and   in the right-handed sense.

In terms of the orthonormal basis  , the cross product can be written in the form of a determinant

 

In index notation, the cross product can be written as

 

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

Identities from Vector AlgebraEdit

Some useful vector identities are given below.

  1.  .
  2.  .
  3.  
  4.  
  5.  
  6.  
  7.  

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

Introduction · Vector Calculus