Dot Product of Plane Vectors

1. Definition of the Dot Product
   Let be the angle between two non-zero vectors and . The dot product is defined as follows:
1. If , then:
   
2. If , then:
   
3. If or , then .


2. Dot Product in Terms of Components
   For two vectors and :
   
   
3. Properties of the Dot Product
   For three vectors , , and :
1. Commutative Property:
2. Distributive Property:
   
   
3. Scalar Multiplication: (where is a real number)

Angle Between Two Plane Vectors

Let be the angle (where ) between two non-zero vectors and :

1. If , then:
2. If , then

Orthogonality and Parallelism of Two Plane Vectors

For two non-zero vectors and :