Zero Vector

If we add two vectors  and , we get a vector. Suppose the vectors  and  have equal magnitudes but opposite directions. What is the vector  +  ? The magnitude of the vector will be zero. For mathematical consistency, it is appropriate to have a vector of zero magnitude but it has a little significance in physics. This vector is called Zero vector. The direction of zero vector is indeterminate. We can write this vector as zero with a line over it exactly as a vector is represented.

Dot Product & Cross Product Of Two Vectors

The dot product (also called scalar product) of two vectors is defined as
  . = ab cosθ
where a and b are the magnitudes of  and  respectively and θ is the angle between them.


The cross product (also called vector product) of two vectors  and , denoted by  *  and is itself a vector. The magnitude of this vector is
 |  *  |= ab sinθ

Subtraction Of Vectors

Let  and  be two vectors. We define  -  as the sum of the vector  and the vector (-) . To subtract  from , invert the direction of  and add to .

Addition Of Vectors

Vectors can be added by various rules.

TRIANGLE LAW OF VECTOR ADDITION


Given two vectors  and  , their sum or resultant written as ( + )  is a vector obtained by first bringing the initial point of to the terminal point of and then joining the initial point of to the terminal point ofgiving a consistent direction by completing the triangle OAB  direction by completing the triangle OAB                        Note that addition is commutative


PARALLELOGRAM LAW OF VECTOR ADDITION

The sum can also be obtained by bringing the initial points of and together and then completing the parallelogram OACB


Note that addition is commutative
Also, + ( + ) = ( +  ) +  i.e. the addition of vectors obeys the associative law. If   and   are collinear, their sum is still obtained in the same manner although we do not have a triangle or a parallelogram in this case.


POLYGON LAW OF VECTOR ADDITION


For adding more than two vectors, we have a polygon law of addition which is just an extension of the triangle law.



A consequence of this is that, if the terminus of the last vector coincides with the initial point of the first vector, the sum of the vectors is . To obtain  -  (difference of two vectors), perform addition of  and (-).


Also, ; ;

Properties of Vector Addition:

 +  =  +  vector addition is commutative

 +  ( + ) +  vector addition is associative

 +  = 
 + (-) = 
(k₁ + k₂) = k₁ + k₂
k ( + ) = k + k