If STP ( Scalar Triple Product ) of 3 vectors a , b and c is negative then it follows left handed system.

Question

I was revisiting through my class notes and found this, there ain't any detailed explanation in my notes. So, I'm kinda confused what does the left-hand system mean? And I don't remember ever using my left hand to find the direction of a cross b. Could you please provide an example for the same?

Answer 1

Consider your thumb the $x$ axis, forefinger the $y$ axis, and middle finger the $z$ axis. You can extend them so that they are all perpendicular.

There are two handednesses in a 3D coordinate system axes. If you rotate such a coordinate system, it will always match either the one shown by your left hand, or the one shown by your right hand.

In particular, if $x$ axis points right, and $y$ axis points up, then the $z$ axis can either point away (left handed) or towards the viewer (right handed coordinate system).

Similarly, in a 2D system, if $x$ axis points right, then $y$ axis can point either up (right handed) or down (left handed). The 2D analog of the cross product between two vectors is then positive if the two vectors defining a parallelogram are in the same order (second in the same direction from the first, as $y$ axis is from $x$ axis).

Scalar triple product in 3D, and the 2D analog of cross product, do not tell anything about the coordinate system handedness. It only tells you whether the vectors involved have the same handedness as the coordinate system.

For example, consider $\hat{u} = (1, 0, 0)$, $\hat{v} = (0, 1, 0)$, and $\hat{w} = (0, 0, 1)$. Their triple product $\hat{u} \cdot (\hat{v} \times \hat{w}) = (\hat{u} \times \hat{v}) \cdot \hat{w} = \hat{u}\hat{v}\hat{w} = 1$. However, if we reorder them so that no rotation can put them in the same order as the coordinate axes in that coordinate system, say $\hat{v}$, $\hat{u}$, $\hat{w}$, then their triple product is $-1$.

In a more general sense, the sign of the triple product tells us whether the three vectors defining a parallelepiped have the same handedness as the coordinate system or not. Since it is usually assumed that the coordinate system itself is right-handed –– it's just a convention ––, a negative sign indicates the three vectors are in left-handed order. However, if the coordinate system itself is left-handed, then a negative triple product indicates the three vectors defining the parallelepiped are in right-handed order. The magnitude of the triple product is the volume of said parallelepiped.