I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to
this,
sense is specified by two points on a line parallel to a vector. Orientation is specified by the relationship between the vector and given reference lines (which I'm interpreting to be some basis).
However, these two definitions seem to be synonymous with direction. How do these 3 terms differ?
Best Answer
For the purposes of this answer two nonzero vectors ${\bf x}$, ${\bf y}\in{\mathbb R}^d$ are considered as equivalent if there is a $\lambda>0$ such that ${\bf y}=\lambda\,{\bf x}$. An equivalence class is called a direction, and two vectors belonging to the same equivalence class are said to point into the same direction. The unit sphere $S^{d-1}\subset{\mathbb R}^d$ is a set of representatives for this equivalence relation.
In a one-dimensional setting one has just two directions which then are called senses. They are represented by the two points $1$ and $-1$ making up $S^0\subset{\mathbb R}^1$.
The notion of orientation refers to bases of $d$-dimensional real vector spaces $V$. Two bases $(a_i)_{1\leq i\leq d}$ and $(b_i)_{1\leq i\leq d}$ of $V$ are equally oriented when the matrix $T$ relating them has positive determinant. There are exactly two equivalence classes. When there is a distinguished basis of $V$ (e.g. the standard basis $(e_i)_{1\leq i\leq d}$ of ${\mathbb R}^d$) its orientation is usually considered the positive orientation.
An example: When a hyperplane $H\subset V$, $\>0\in H$, is given then a chosen positive orientation in $V$ induces an orientation of $H$ only after a positive normal vector ${\bf n}\perp H$ has been selected. A basis $(a_i)_{1\leq i\leq d-1}$ of $H$ is then positively oriented if $({\bf a}_1,\ldots, {\bf a}_{d-1},{\bf n})$ is a positively oriented basis of $V$.