When I was studying about vectors and their use in physics, I found something called zero vector. My physics textbook says it is
A vector whose initial and terminal points coincide is called zero vector, it has zero magnitude but an arbitrary direction, i.e. it cannot be assigned a direction.
My question is What is the significance of this zero vector?
For example, If the force acting on a body has no magnitude then is there any meaning/significance to say that the force has a direction? It can also be said that a vector +zero vector = same vector, then what?? Zero vector has bring no change in the vector. I m totally confused regarding the role of this zero vector in mathematics/physics. Please help. Thanks
One property of the zero vector is that it is known as the "additive identity". Just like multiplying a number by 1 is the multiplicative identity, multiplying any number by 1 gets the same number back, adding $\vec{0}$ gets the same vector back. That makes it a unique element in any vector space, and it is the one vector every vector space is required to have. In fact, the zero dimensional vector space consists of the set $\left\{\vec{0}\right\}$.
As for thinking about magnitude and direction, that requires a vector space to have another structure that allows you to calculate the length of a vector. That structure is called a metric. The general definition of a vector's direction is the unit vector that points in the same direction as the vector. In notation, a vector's direction is given by: $$\hat{n} = \frac{\vec{v}}{|\vec{v}|}.$$ For the zero vector, you get $0/0$ for each component, and thus, no definable direction for the zero vector.