The above diagram shows the simple case of rotation about the origin, if we In 2 dimensions there is one degree of freedom, in 3 dimensions In order to investigate the issues, consider the simpler case of rotations ![]() When we are dealing with torque, inertia etc. So for angular velocities (w=da/dt) vector additionĬan be used, however we need to be very careful (for example rotational effects This is because as the angles get smaller then combining rotations begins toĪpproximate vector addition. Vectors like this are useful for representing some types of rotational quantities. To work out the effect of combining rotations. We cant find the result of applying subsequent rotations by just adding vectorsĪnd order of applying the rotations is important, we have to use different types However, in the rotational case we cannot make these assumptions, In neither case does this produce a component The order that I do these operations is not important, We must be careful when combining rotations which are in a different plane, in this case we cannot combine them by adding the bivectors, we need other notations to do this.įor instance, in the case of linear translations, I could move an object upīy 3 units (represented by ) and then move the object to the right byĤ units (represented by ). It just happens that in 3 dimensional vector space that bivectors also have three dimensions and therefore 3D rotations have 3 degrees of freedom. For more information about rotations in higher dimensions see this page. The dimension of the bivector also gives the degree of freedom of rotations in the given dimension. A bivector is the result of multiplying two vectors, the following table shows the dimension of the bivector produced by the multiplication of vectors of a given dimensions. The plane can be defined by a bivector (see directed area box on right of this page). Any rotation can be represented by projecting the object onto a 2-dimentional plane and then rotating it through an angle. Rotations in a higher number of dimensions get more complicated. Rotations in two dimensions are relatively easy, we can represent the rotation angle by a single scalar quantity, rotations can be combined by adding and subtracting the angles. You can specify the position of a point using polarĬoordinates, this is covered separately from the topic of rotations.
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