Why vector form is convenient?

[manimaran1605]

I read that a point relative to origin in 3D coordinate system is conveniently specified by the vector form, why?

 

[arildno]

Because
a) the vectorial description contains all the info required to identify the point, and nothing superfluous
b) The manner in which we write vectors makes it easy for us to treat them as a type of numbers; that is, we easily see how to add, and for that matter (sort of) multiply them together, and also how to interpret, in a geometric sense, what such algebraic operations "really" means.

 

[HallsofIvy]

Because there are many different points the same distance from the origin (or anyone point) of a coordinate system so a single number will not suffice to identify it. Equivalently, it take 3 numbers to identify everypoint in a 3D coordinate system (pretty much the definition of "3D") and it is convenient to arrange those numbers in an array. It is the fact, from geometry of similar triangles, that the coordinates $(x_0, y_0, z_0)$ of a point exactly 1/2 way between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are $(x_0, y_0, z_0)= ((x_1+ x_2)/2, (y_1+ y_2)/2, (z_1+ z_2)/2)$ that means that "scalar multiplication" of vectors is convenient (in fact, "scalar multiplication" and the whole idea of vectors was created to simplify that).

(Arildno got in two minutes before me! And we are saying essentially the same thing.)

 

[manimaran1605]

got it, Thank you both for clarifying my doubt

 

[CellCoree]

Vector form is convenient for specifying a point relative to the origin in a 3D coordinate system because it allows for a concise and efficient representation of the point's position in space. Unlike other forms, such as Cartesian coordinates, which require three separate values to define a point, vector form only requires a single vector with three components. This makes it easier to perform mathematical operations and transformations on the point, as well as to visualize its location in relation to other points in the coordinate system. Additionally, vector form is useful in many scientific and engineering applications, such as physics and computer graphics, where precise and efficient representation of points in space is crucial. Overall, vector form offers a convenient and versatile way to specify points in 3D coordinate systems.