Torque the vector cross product, p is the particle's linear momentum, and r is the displacement vector from the origin (the origin is assumed to be a fixed location anywhere in space). The time-derivative of this is: This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force formula_17 (whether or not mass is constant) and the definition of velocity formula_18 The cross product of momentum formula_20 with its associated velocity formula_21 is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque τ
Angular velocity origin: The relation between this linear map and the angular velocity pseudovector formula_74 is the following. Because "W" is the derivative of an orthogonal transformation, the bilinear form is skew-symmetric. Thus we can apply the fact of exterior algebra that there is a unique linear form formula_76 on formula_77 that where formula_79 is the exterior product of formula_80 and formula_81. Taking the sharp "L" of "L" we get Introducing formula_83, as the Hodge dual of "L", and applying the definition of the Hodge dual twice supposing that the preferred unit 3-vector is formula_84 where by definition. Because formula_81 is an