# Givens rotation and right-hand rule

If $G(i, j, \theta)$ denotes a Givens rotation in the $(i, j)$ plane of $\theta$ radians, then:

The product $G(i, j, θ)x$ represents a counterclockwise rotation of the vector $x$ in the $(i, j)$ plane of $\theta$ radians, hence the name Givens rotation.

Then later it says

... a Givens rotation ... does not necessarily respect the right-hand rule ...

Well as I know the right-hand rule defines rotations in counter-clockwise order.

Now here is the confusion: if $G(i, j, \theta)x$ is a rotation of the vector $x$ in counter-clockwise order, and if right-hand rule also defines rotation in counter-clockwise order, how is it possible that Givens rotation doesn't necessarily respect the right-hand rule?

• What do you mean by "right-hand rule"? I mean, a rotation is a rotation. – Martin Argerami Oct 7 '16 at 4:03
• Givens rotation matrices in 3D and en.wikipedia.org/wiki/Right-hand_rule#Rotation – plasmacel Oct 7 '16 at 4:24
• Exactly. The right-hand rule tells you the direction of the normal vector in certain cases where you need it. It is not intrinsic to the rotation in any sense. As I said, a rotation is a rotation. – Martin Argerami Oct 7 '16 at 4:30
• I think you don't understand the question. There is a direction of positive rotation. In the right-hand rule, the positive direction of rotation is counter-clockwise to the normal of the plane of rotation. The Givens rotation doesn't follow this convention. If you compare a basic 3D rotation matrix which rotates on the XZ plane (Y axis), and a Givens rotation which rotates about the XZ plane, they rotate in opposite directions. However for XY and ZY planes they are the same. – plasmacel Oct 7 '16 at 4:35
• Compare the two $R_Y(\theta)$ at en.wikipedia.org/wiki/Givens_rotation#Dimension_3 – plasmacel Oct 7 '16 at 4:43