Areal velocity equal to the magnitude of the cross product of vectors "AB" and "AC". This area can also be viewed as a vector with this magnitude, pointing in a direction perpendicular to the parallelogram; this vector is the cross product itself: Hence The areal velocity is this vector area divided by Δ"t" in the limit that Δ"t" becomes vanishingly small: But, formula_4 is the velocity vector formula_5 of the moving particle, so that On the other hand, the angular momentum of the particle is and hence the angular momentum equals 2"m" times the areal velocity. Conservation of areal velocity is a
Dot product identity, also known as "Lagrange's formula" may be remembered as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula finds application in simplifying vector calculations in physics. In physics, vector magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. Examples include: For vectors with complex entries, using the given definition of the dot