in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products. Physics Vectors have many uses in physics and other sciences. Length and units In abstract vector spaces, the length of the arrow depends on a dimensionless scale. If it represents, for example, a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1 m:50 N and 1:250 respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. Vector-valued functions Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter t. For instance, if r represents the position vector of a particle, then r(t) gives a parametric representation of the trajectory of the particle. Vector-valued functions can be differentiated and integrated by differentiating or integrating the components of the vector, and many of the familiar rules from calculus continue to hold for the derivative and integral of vector-valued functions. Position, velocity and acceleration The position of a point x = (x1, x2, x3) in three-dimensional space can be represented as a position vector whose base point is the origin The position vector has dimensions of length. Given two points x = (x1, x2, x3), y = (y1, y2, y3) their displacement is a vector which specifies the position of y relative to x. The length of this vector gives the straight-line distance from x to y. Displacement has the dimensions of length. The velocity v of a point or particle is a vector, its length gives the speed. For constant velocity the position at time t will be where x0 is the position at time t = 0. Velocity is the time derivative of position. Its dimensions are length/time. Acceleration a of a point is vector which is the time derivative of velocity. Its dimensions are length/time2. Force, energy, work Force is a vector with dimensions of mass×length/time2 and Newton's second law is the scalar