Vector algebra
Encyclopedia
In mathematics, vector algebra may mean:
- Linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. - The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
of dot productDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
and especially cross productCross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
. In this sense, vector algebra is contrasted with geometric algebraGeometric algebraGeometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...
, which provides an alternative generalization to higher dimensions. - Original vector algebras of the nineteenth century like quaternionQuaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s, tessarineTessarineIn mathematics, a tessarine is a hypercomplex number of the formt = w + x i + y j + z k, \quad w, x, y, z \in Rwhere i j = j i = k, \quad i^2 = -1, \quad j^2 = +1 .The tessarines are best known for their subalgebra of real tessarines t = w + y j \ ,...
s, or coquaternionCoquaternionIn abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a...
s, each of which has its own productProduct (mathematics)In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
. The vector algebras biquaternions and hyperbolic quaternionHyperbolic quaternionIn the abstract algebra of algebras over a field, the hyperbolic quaternionq = a + bi + cj + dk, \quad a,b,c,d \in R \!is a mutated quaternion wherei^2 = j^2 = k^2 = +1 \! instead of the usual −1....
s enabled the revolution in physics called special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
by providing mathematical models.