Scalar Multiplication of Vectors

Scalar Multiplication of Vectors

The product of a real number and a vector is called the scalar multiplication of the vector .

1. When :
1. If , the vector has the same direction as and a magnitude of .
2. If , then .
3. If , the vector has the opposite direction to and a magnitude of .
   
   
2. When , .

Properties of Scalar Multiplication

For two real numbers and and two vectors and :

1. Associative Property
   
2. Distributive Property
   
   

If , then is a vector with the same direction as and a magnitude of , i.e., is a unit vector in the direction of .

Parallel Vectors

1. Parallel Vectors
   Definition of Parallel Vectors:
   Two non-zero vectors and are said to be parallel if their directions are either the same or opposite. This is denoted as:
   
   
2. Condition for Two Vectors to be Parallel
   Two non-zero vectors and are parallel if and only if there exists a non-zero scalar such that:
   
   
3. Condition for Three Points to be Collinear:
   For three distinct points , , and , the points are collinear if and only if there exists a non-zero scalar such that:
   This implies that , and the three points , , and lie on a straight line.
   Conversely, if three points , , and are collinear, then there exists a non-zero scalar such that: