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
Dot product representation of a graph Dot product representation of a graph A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot product representation. Let "G" be a graph with vertex set "V". Let "F" be a field, and "f" a function from "V" to "F" such that "xy" is an edge of "G" if and only if "f"("x")·"f"("y") ≥ "t". This is the dot product representation of "G". The number "t" is called the dot product threshold, and the smallest possible value of "k" is