[Math] vectors : What is the meaning of right handed system

vectors

In a book, I read the following

"Having a fixed frame of reference, we introduce three unit vectors $\bf i, \ j,\ k$ along the directions of the $X, Y, Z$ axis respectively $\dots$ Note that every two distinct vectors in the set $\{\bf i$, $\bf j$, $\bf k$$\}$ are orthogonal. Therefore the system ($\bf i$, $\bf j$, $\bf k$) is called ${\bf orthogonal}$. Further, since all its elements are unit vectors, it is said to be an ${\bf orthonormal\ system}$. Note that $\bf i$ $\times \ \bf j$ is a unit vector perpendicular to both $\bf i$ and $\bf j$. Therefore $\bf i\ \times\ \bf j$ is either $\bf k$ or $- \bf k$. The first possibility holds when the system is right-handed, the second when it is left-handed. In a right-handed orthonormal system, we have the following cyclic formulas
$$
\bf i\ \times \ j = k, \quad j\ \times \ k = i, \quad k\ \times \ i = j
$$
$\dots$"$\\ \\$

My doubts are

  1. Is the author assuming that the vectors $\bf i,\ j, \ k$ are along the positive direction of axes?

2.What is the meaning of right-handed system? It must have some meaning which is independent of $\bf i,\ j,\ k$.

3.Why have we made the concept of 'right-handed system' and 'left-handed system'? Is there any advantage of one on another?

Best Answer

  1. Yes.
  2. An orthonormal system $({\bf e}_1, {\bf e}_2, {\bf e}_3)$ is called right-handed if ${\bf e}_1 \times {\bf e}_2 = {\bf e}_3$ (equivalently, the same expression with $1, 2, 3$ permuted cyclically). We can extend the notion to general bases $({\bf f}_1, {\bf f}_2, {\bf f}_3)$ of $\Bbb R^3$ by declaring a basis to right-handed if $({\bf f}_1 \times {\bf f}_2) \cdot {\bf f}_3 > 0$ (again, we can freely cyclically permute the indices in this definition).
  3. There is no intrinsic advantage to either convention. It is advantageous, however, for everyone to use the same convention, and we generally use right-handedness.
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