Sorry for asking a very common question. For now this is a question which make me depressed for about an year. As a mathematics student I studied vectors very axiomatically which makes me feel depressed. Now my question is what is a vector. Please don't say that vectors are quantities having both magnitude and direction. Please explain that.
What does one mean by the word position vector?
Do the vector position vector and the vector with same magnitude and direction is same? (refer to picture)
Do $1$ and $2$ represent the same vectors?
The vector $2\mathbf{i}+3\mathbf{j}$ is represent by an arrow pointing the point $(2,3)$ in the plane. But how can we represent the vector $x\mathbf{i}+y\mathbf{j}$ where $x$ and $y$ are variables.
Please suggest me a good book which has a good approach to vectors from the basic. I want to understand the practical difference between the vectors and the statement "each point in a plane is a vector".
Please help me.
Best Answer
You certainly know that numbers could refer to different kind of object. $5$ could be an area of a triangle, could be the number of apples, could be the value in dollar of an object.
The same works for vectors. Vectors are often used to represent force. But you could use a vector to represent a direction, speed or acceleration, or even they can be used to represent a color in RGB system.
In your picture you have the same vectors, but they have different point of application: the first one is applied in $(0,0)$ the second one, somewhere in the third quadrant.
Vectors could also have 3 dimension, or even n dimension. So don't be confused about them: they are used to represent some concept, but actually they are just "numbers", and they can have the meaning you wish.