| • Science | • People | • Locations | • Timeline |
In other words, it maps a pair of vectors to a scalar. When the latter term is used, the inner product of a and b is usually denoted <a, b>; see the article inner product space for a more abstract treatment.
It is defined as
or, using italics to denote the norm of a vector (i.e., x ≡ |x|),
where θ is the angle between the two vectors. Thus, the dot product of two perpendicular vectors is always zero. If a and b are both unit vectors (i.e., of length 1), the dot product simply gives the cosine of the angle between them. Thus, given two vectors, the angle between them can be found by rearranging the above formula:
This can be understood very easily: The first vector is projected onto the second vector (the order does not matter as the dot-product is a commutative) by calculating the dot-product, and afterwards 'normalized' by dividing the obtained scalar value of the numerator through their scalar length. Thus the scalar value of the fraction must be lesser or equal 1 and can be easily translated into a angular value (As the trigonometric functions are really nothing more than taylor approximated functions to achieve a seamless translation table of lengths into angle values and vice versa (arcsin,...). More to that on the sine page).
The dot product is particularly used in the calculation of net force. If b is a unit vector, then the dot product gives the projection of a in direction b. In mechanicsMechanics ( Latin mechanicus from the Greek mechanikos "one skilled in machines") is a variety of specialised sciences pertaining to the functions and routine operations of machines, machine-like devices or objects. When preceded by a qualifier, mechanics, this gives the component of a forceIn physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. The concept appeared first in the second law of motion of classical mechanics. It is usually expressed by the equation F m · a where F is the force, in that direction.
WorkWork (abbreviated W is the energy transferred in applying force over a distance. Work can be calculated from the formula: : : where F is the force. s is the position. Readers not familiar with vectors and calculus please see "Simpler Formulae" below. is the dot product of force and displacement.The definition has the following consequences. The dot product is commutative
Two non-zero vectors a and b are perpendicularPerpendicular is a geometric term that may be used as a noun or adjective. The fundamental meanings pertain to the position of straight lines relative to one another, in which two lines are positioned at an angle of ninety degrees, which is defined as a r if and only if a · b = 0. The dot product is bilinear
From these it follows directly that the dot product of two vectors a = [a1 a2 a3] and
b = [b1 b2 b3] given in coordinates can be computed particularly easily as
or, using matrix multiplicationThis article gives an overview of the various ways to multiply matrices. The Einstein notation is used throughout. Ordinary matrix product By far the most important way to multiply matrices is the usual matrix multiplication. It is defined between two mat and treating the vectors as 1-by-3 matrices,
where bT denotes the transpose of the matrix b.
The dot product satisfies all the axioms of an inner product. In an abstract vector space, the notion of angle between the elements of the space can be defined in terms of the inner product.