a unit vector perpendicular to the plane of angular displacement, a scalar angular speed results, where and where is the perpendicular component of the motion, as above. The two-dimensional scalar equations of the previous section can thus be given direction: and for circular motion, where all of the motion is perpendicular to the radius . In the spherical coordinate system the angular momentum vector expresses as Angular momentum in any number of dimensions Angular momentum can be defined in terms of the cross product only in three dimensions. Defining it as the bivector , where is the exterior product, is valid in any number of dimensions. This exterior product is equivalent to an antisymmetric tensor of degree 2, which also applies in any number of dimensions. Namely, if is a position vector and is the linear momentum vector, then we can define In the general case of summed angular momenta from multiple particles, this antisymmetric tensor has independent components (degrees of freedom), where is the number of dimensions. In the usual three-dimensional case it has 3 independent components, which allows us to identify it with a 3 dimensional pseudovector . The components of this vector relate to the components of the rank 2 tensor as follows: Analogy to linear momentum Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape. Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product, is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment