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Comparison of vector algebra and geometric algebra
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Comparison of vector algebra and geometric algebra : ウィキペディア英語版
Comparison of vector algebra and geometric algebra
Vector algebra and geometric algebra are alternative approaches to providing additional algebraic structures on vector spaces, with geometric interpretations, particularly vector fields in multivariable calculus and applications in mathematical physics.
Vector algebra is specific to Euclidean 3-space, while geometric algebra uses multilinear algebra and applies in all dimensions and signatures, notably 3+1 spacetime as well as 2 dimensions. They are mathematically equivalent in 3 dimensions, though the approaches differ. Vector algebra is more widely used in elementary multivariable calculus, while geometric algebra is used in some more advanced treatments, and is proposed for elementary use as well. In advanced mathematics, particularly differential geometry, neither is widely used, with differential forms being far more widely used.
== Basic concepts and operations ==
In vector algebra the basic objects are scalars and vectors, and the operations (beyond the vector space operations of scalar multiplication and vector addition) are the dot (or scalar) product and the cross product ×.
In geometric algebra the basic objects are multivectors (scalars are 0-vectors, vectors are 1-vectors, etc.), and the operations include the Clifford product (here called "geometric product") and the exterior product. The dot product/inner product/scalar product is defined on 1-vectors, and allows the geometric product to be expressed as the sum of the inner product and the exterior product when multiplying 1-vectors.
A distinguishing feature is that vector algebra uses the cross product, while geometric algebra uses the exterior product (and the geometric product). More subtly, geometric algebra in Euclidean 3-space distinguishes 0-vectors, 1-vectors, 2-vectors, and 3-vectors, while elementary vector algebra identifies 1-vectors and 2-vectors (as vectors) and 0-vectors and 3-vectors (as scalars), though more advanced vector algebra distinguishes these as scalars, vectors, pseudovectors, and pseudoscalars. Unlike vector algebra, geometric algebra includes sums of ''k''-vectors of differing ''k''.
The cross product does not generalize to dimensions other than 3 (as a product of two vectors, yielding a third vector), and in higher dimensions not all ''k''-vectors can be identified with vectors or scalars. By contrast, the exterior product (and geometric product) is defined uniformly for all dimensions and signatures, and multivectors are closed under these operations.

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