Is it necessary that two vectors will always be coplanar?

Question

Is it necessary that in 3 dimensions, two vectors will always be coplanar ? My teacher told the class that in 3 dimensions two vectors are always coplanar. But what if we consider two non parallel and non intersecting vectors ? like the ones parallel to 2 skew lines how can these be co planar ?

Another example is consider a cube. Now take one vector that is along the face diagonal of the upper face of the cube and another one that is along the face diagonal of the bottom face (the diagonal that is not in the same direction as the other one). These 2 are not coplanar as well.

Answer 1

I think you, your professor and the other answer mixed different definition of vectors used in physics and in mathematics - which is understandable since this forum is about mathematics.

In physics is often useful to use free vectors (or Euclidian vectors) which have a direction and magnitude, and localized vectors (or bounded vectors or affine vectors) which also have a point of application. This way, localized vectors are defined by two points while free vectors are defined by just one point. In fact, free vectors are like localized vectors with a common start. A summary of why physicists often use localized vectors may be found on https://physics.stackexchange.com/questions/139824/why-is-force-a-localized-vector-and-not-a-free-vector.

On the other hand, in mathematics, when talking about vectors and vector spaces, we nearly always mean free vectors.

Then, it's true that two free vectors are always coplanar (as the other answer says), but two localized vectors may not be coplanar (as the question says).

Answer 2

You're confusing vectors with line segments and linear spaces with affine spaces.

When you take the example of the cube, you describe line segments. Those are indeed not coplanar. But those are also not vectors.

If you want to visualize (not a good mathematical word by the way...) coplanar vectors, you have to position the origin of those vectors at a common point. If the other vertices of those vectors and the common points are coplanar, then you can say that those vectors are coplanar.

When you apply this logic, you'll get the conclusion that any two vectors are always coplanar. Which by the way is not only true in dimension $3$ but for any dimension $n \ge 2$.