Say i have two vectors A and B. Mathematically they are same if they have same magnitude and direction.
So, say if someone asks me to draw a "vector" of 5 magnitude with 45 degree angle with x-axis.. i can draw infinite such vectors on the graph.
My question is, what is the mathematical term, say if i want to have a special vector having fixed end points. So that if i say draw " _____ " that has magnitude 5, with 45 degree, and starts from [x,y] and ends at [x2,y2] ?
On
I have sometimes heard this called a bound vector (as opposed to a free vector). But most people in the comments are right - this isn't a standard term, and terms like this are normally used in very specific, technical uses or as teaching aids. In most standard mathematics, vectors don't have start- and end-points - that is, regardless of where you draw it, if it has the same magnitude and direction, it's the same vector. All vectors are free.
A vector, by definition, is only a magnitude and a direction. If someone asks you to draw a given vector on paper, you can draw an infinite number of them, but they are just instances of the same vector.
By analogy, suppose someone asked you to write the number $6$. No matter where you write it on the paper, it is still a $6$.
I believe the term you're looking for is a directed line segment, which is distinct from a vector. A directed line segment can be defined in one of two ways:
parametric equations: $$\begin{align} x(t)&= at+b \\ y(t)&= ct+d \\ t_1&\le{t}\le{t_2} \end{align}$$
or basic Euclidean coordinate functions: $$y=mx+b,x_1\le{x}\le{x_2}$$