# Vector Algebra

Vector addition and subtraction satisfy the following properties:
1. Commutivity The order of adding vectors does not matter.

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 ,

4. Inverse Element for Vector Addition For every vector , there is a unique inverse vector
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such that
This means that the vector  has the same magnitude as , i.e.,  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,
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# Vector Multiplication or Product

When two vectors  and  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  and  written as,  is defined as,

where,  is the smaller angle between  and  and  represent the magnitude of  and , respectively.
1. Properties of Dot Product
1. The first property is that the dot product is commutative.
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1. The second property involves the dot product between a vector  (where c is a scalar) and a vector .
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1. The third property involves the dot product between the sum of two vectors  and  with a vector .
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This shows that the dot product is distributive.â€‹
1. Since the dot product is commutative, similar relations are given, e.g.,
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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,
1. Application of Dot Product

One major application of dot product is to find the work done by a force  for a displacement of , given as,

# Vector Product, or Cross Product

The cross product of two vectors  and , written as , is defined as,

where  is the unit vector normal to the plane containing  and .

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.