Angle Between Vectors And Magnitude of Vectors

Angle Between Vectors And Magnitude of Vectors
The angle between two vectors is represented by the smaller of the two angle between the vectors when they are placed tail to tail by displacing either of the vectors parallel to it self or simply when they are co-initial or co-terminal.
Ex: The angle between \(\overrightarrow A \) and \(\overrightarrow B\) is correctly represented in the following figures

a)If the angle between \(\overrightarrow A \) and \(\overrightarrow B\) is \(\theta\) , then the angle between \(\overrightarrow A \) and \(K\overrightarrow B\) is also \(\theta\). Where `K’ is a positive constant.    

b)If the angle between \(\overrightarrow A \) and \(\overrightarrow B\) is \(\theta\) , then the angle between \(\overrightarrow A \) and \(-K\overrightarrow B\) is  \(180-\theta\). Where `K’ is a positive constant.           

c) Angle between collinear vectors is always zero or 180⁰    

Note: Angle between vectors in anticlockwise direction is taken as positive and clockwise direction as negative
Basic addition of vectors
    To add two vectors, one vector is drawn to scale and the tail of second vector is made to coincide with the head of first vector without changing its orientation.
    
    Then the line joining the tail of first vector and the head of last vector represents the resultant both in mangnitude and direction.
Laws of Vector Addition:
Vector addition follows commutative,associative and distributive laws.
        a) Commutative law : \(\overline A {\text{ + }}\overline B {\text{ = }}\overline B {\text{ + }}\overline A {\text{ }}\)
        b) Associative law : \(\overline A {\text{ + }}\left( {\overline B + \overline C } \right){\text{ = }}\left( {\overline A {\text{ + }}\overline B {\text{ }}} \right) + \overline C \)
        c) Distributive law :\({\text{ m}}\left( {\overline A {\text{   +  }}\overline B {\text{ }}} \right) = {\text{m}}\overline A  + m\overline B \) where m is a scalar
Subtraction of a vector from another vector:
    The subtraction of one vector from the other is a special case of addition of two vectors in which one vector is compounded with the negative of another.  
\(\overline a - \overline b = \overline a + \left( { - \overline b } \right)\)