What are the $\{x,y,z\}$ values of a vector?

Question

A vector is often described as $\{x,y,z\}$ similarly to a $3$D point's cartesian coordiantes in CAD tools which is quite confusing. What are the $x$, $y$ and $z$ values in the case of a vector?

Answer 1

First, $\{x,y,z\}$ denotes a set of things with no order. Coordinates are typically defined as ordered sets. So they're denoted $(x,y,z)$ instead.

Mathematically speaking a vector is defined as an abstract object living in a space. Only when you choose a "basis" for the space, we can then attach a "coordinate" to the vector. In this case, the coordinates could be $(x,y,z),$ or "length and angles from the axes," or "length and direction" etc.

To answer your question, I assume you're working in a 3D space equipped with the Euclidean norm, and the standard basis. Let the your vector be $v = (x, y, z).$ Think of $v$ as the oriented line segment (pointy arrow) from the origin $(0,0,0)$ to the point $(x,y,z).$ The amplitude of the vector (aka length) is $$\| v \| = \sqrt{x^2 + y^2 + z^2}.$$ The direction of $v$ would be the unit vector in the direction of $(x,y,z).$ That is, $$\widehat v = \frac{v}{\| v \|} = ( \frac{x}{ \sqrt{x^2 + y^2 + z^2} }, \frac{y}{ \sqrt{x^2 + y^2 + z^2} }, \frac{z}{ \sqrt{x^2 + y^2 + z^2} } ).$$

Here is a not pretty picture of the situation in 2D