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Vector subtraction is performed by adding the scalar multiple of −1 with the second vector operand to the first vector operand. This can be represented by the use of the minus sign as an operator. The difference between two vectors u and v can be represented in either of the following fashions: +
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
R n understood as an affine space is the same space, where R n as a vector space acts by translations. Conversely, a vector has to be understood as a " difference between two points", usually illustrated by a directed line segment connecting two points.
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". [4] It was first used by 18th century astronomers investigating planetary revolution around the Sun. [5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B.
Consider the real Euclidean n-dimensional space, that is R n = R × R × ... × R (n times) where R is the set of real numbers and × denotes the Cartesian product, which is a vector space. The coordinates of this space can be denoted by: x = (x 1, x 2,...,x n). Since this is a vector (an element of the vector space), it can be written as:
Position vector r is a point to calculate the electric field; r′ is a point in the charged object. Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues. [citation needed] Electric transport
If R is a ring, we can define the opposite ring R op, which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in R op. Any left R-module M can then be seen to be a right module over R op, and any right module over R can be considered a left module over R op.