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Vector Algebra

Vector addition and subtraction satisfy the following properties:
  1. Commutivity The order of adding vectors does not matter.
    Description: 23717.png
  2. Associativity When adding three vectors, it does not matter which two we start with:
    Description: 23727.png
  3. Identity Element for Vector Addition There is a unique vector, 0, that acts as an identity element for vector addition.
    This means that for all vectors Description: 23736.png,
    Description: 23743.png 
  4. Inverse Element for Vector Addition For every vector Description: 23751.png, there is a unique inverse vector
Description: 23759.png 
such that  Description: 23768.png
This means that the vector Description: 23778.png has the same magnitude as Description: 23787.png, i.e., Description: 23794.png but they point in opposite directions.
  1. Distributive Law for Vector Addition Vector addition satisfies a distributive law for multiplication by a number.
    Let c be a real number. Then,
Description: 23802.png

Vector Multiplication or Product

When two vectors Description: 23810.png and Description: 23819.png are multiplied, the result may be a scalar or a vector depending on how they are multiplied. There are two types of vector multiplication:
  1. Scalar product, or dot product
  2. Vector product, or cross product

Scalar Product or Dot Product

The scalar product, or dot product, of two vectors Description: 23836.png and Description: 23843.png written as, Description: 23852.png is defined as,
Description: 23861.png
where, Description: 23880.png is the smaller angle between Description: 23887.png and Description: 23896.pngDescription: 23903.png and Description: 23912.png represent the magnitude of Description: 23923.png and Description: 23930.png, respectively.
  1. Properties of Dot Product
  1. The first property is that the dot product is commutative.
Description: 23982.png
  1. The second property involves the dot product between a vector Description: 23990.png (where c is a scalar) and a vector Description: 23997.png.
Description: 24007.png
  1. The third property involves the dot product between the sum of two vectors Description: 24016.png and Description: 24023.png with a vector Description: 24030.png.
Description: 24038.png
This shows that the dot product is distributive.
  1. Since the dot product is commutative, similar relations are given, e.g.,
Description: 24047.png
Description: 24057.png
  1. Vector Decomposition and the Dot Product
    We now develop an algebraic expression for the dot product in terms of components. We choose a Cartesian coordinate system with the two vectors having component vectors as,
Description: 24269.png
  1. Application of Dot Product
    One major application of dot product is to find the work done by a force Description: 24457.png for a displacement of Description: 24549.png, given as,
Description: 24708.png

Vector Product, or Cross Product

The cross product of two vectors Description: 25453.png and Description: 25550.png, written as Description: 25642.png, is defined as,
Description: 25734.png
where Description: 25919.png is the unit vector normal to the plane containing Description: 26011.png and Description: 26105.png.
The vector multiplication is called cross product due to the cross sign. It is also called vector product because the result is a vector.
The direction of the cross product is obtained from a common rule, called right-hand rule.

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