# Vector Algebra

Vector addition and subtraction satisfy the following properties:

**Commutivity**The order of adding vectors does not matter.**Associativity**When adding three vectors, it does not matter which two we start with:**Identity Element for Vector Addition**There is a unique vector, 0, that acts as an identity element for vector addition.**Inverse Element for Vector Addition**For every vector , there is a unique inverse vector

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such that

such that

This means that the vector has the same magnitude as , i.e., but they point in opposite directions.

**Distributive Law for Vector Addition**Vector addition satisfies a distributive law for multiplication by a number.*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:

- Scalar product, or dot product
- 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,**Properties of Dot Product**

- The first property is that the dot product is commutative.

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- The second property involves the dot product between a vector (where c is a scalar) and a vector .

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- 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€‹

- Since the dot product is commutative, similar relations are given, e.g.,

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**Vector Decomposition and the Dot Product**

**Application of Dot Product**

# 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*.