Vector and vector quantity I am quite confused about vectors and vector quantity. Somewhere I find they are used interchangeably and somewhere, they have different meanings. So, I wish to know whether there is a difference between vector and vector quantities or not. Is force a vector or a vector quantity. Is vector a directed line segment or a physical quantity which has both magnitude and direction and add according to vector laws of addition?
Please help me.
 A: Saying that force is a "vector quantity" is usually just a way of saying that force is always represented by a vector. If we were to say that force is a "vector", the implication would be that force was only allowed to ever be one vector. However, no one would look sideways at you if you said that force was a vector; it's a  very pedantic distinction.
A vector is a mathematical object which we can work with in the following ways:

*

*We can add two vectors $\mathbf{v}+\mathbf{w}$ to get another vector, and this addition behaves in a standard way ($\mathbf{v}+\mathbf{w}=\mathbf{w}+\mathbf{v}$, there's a zero vector such that $\mathbf{v}+\mathbf{0}=v$, etc.)

*We can multiply a vector by a scalar and get another vector $a\mathbf{v}=\mathbf{w}$ (these scalars are usually just real numbers, but we can also allow complex numbers and other even more general types of numbers).

*These two operations satisfy the distributive property: $a(\mathbf{v}+\mathbf{w}) = a\mathbf{v}+a\mathbf{w}$ and $(a+b)\mathbf{v} = a\mathbf{v}+b\mathbf{v}$.


This definition then allows you to model a lot of interesting physical quantities, like force. In particular, a particular class of vectors, which we denote by $\mathbb{R}^2$, is represented by arrows in the plane which we add and multiply by scalars in the way you're familar with. It's this set of vectors which we usually use to describe force. Force is a vector quantity, meaning that it is always described by a vector $v$, and the laws pertaining to force always involve vector arithmetic for this reason.
A: A vector is a mathematical object that obeys the properties highlighted in Sasha's answer.
An Euclidean vector (usually just called a vector) is a value with a free direction in space; thus, it can be graphically represented as a directed line segment whose length corresponds to the vector's magnitude.
We call an Euclidean vector a ‘vector quantity’ when wishing to emphasise the physical context; thus, a force is called a vector quantity.
