Vectors in $3$-dimensional space one parallel to $x$ axis and other parallel to $y$ axis should be said parallel or non parallel vectors?

Question

How can we say that two vectors are parallel or not in $3$-dimensional space for example one vector being parallel to $x$ axis and other being parallel to $y$ axis should be parallel or non parallel?

Answer 1

Both vectors will be in the same direction if they are proportional. What I mean is that if you consider $(u_1,u_2,u_3)$ and $(v_1,v_2,v_3)$, they will be parallel if there exists some $\lambda\in\mathbb{R}$ such that $(u_1,u_2,u_3) = \lambda(v_1,v_2,v_3)$.

You can also prove it seeing that the cross product is equal to $0$.

In your example, your vectors are perpendicular (non parallel). To see the perpendicularity you can use the scalar product.

Answer 2

All vectors in a vector space are considered to have the origin $(0,0,0)$ as common point, so obviously two vectors, one parallel to the $x$ axis and the other parallel to the $y$ axis, are orthogonal, as they are mutually orthogonal the two axis.

Maybe that you are thinking at two oriented segments that are parallel to the two axis, but have no common point. In this case we can interpret the two oriented segments as two vectors in an affine space, applied two different points of two skew lines, and we can say that they are two skew vectors.