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Firstly, What I donot understand is the need for vector-valued function. For example, we know that a position vector determines the position of a point in two or three dimensional space and the vector field or vector-valued function assigns this vector. Therefore why we use vector-valued function when the solution (value of x & y) of ordinary function can be represented graphically which also represents a unique point.

Secondly, I can fully imagine the graphical representation of a function in three dimensions (to some extent if the fourth dimension is time) but I cannot imagine a n variable function graphically. What is the logic behind these sort of functions and is it possible to graph these functions and if yes how?

Finally, I am learning about the vector-valued function of more than three dimensions therefore how the vector field assigns a unique vector to the points generated by these vectors (keeping in mind that we have only three units vectors namely i, j and k along x, y and z axes)?

Please answer these questions very comprehensively because I am very confused by these questions.