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Introduction To Vectors

Vectors

Vectors

In mathematics, a vector is a very general concept. This article will deal with vectors in the context of GLScene, where they are represented as a list of numbers. An example of a vector with two entries is v = (5, 7). We call the set∞ containing all possible vectors with two entries for R², where R stands for the real numbers∞ which appear as entries in the vectors. The exponent indicates how many entries each vector contains. Two vectors are equal if and only if their corresponding entries are equal. So the vector (1, 3) is not equal to the vector (3, 1).

Given two vectors u and v, the sum u + v is calculated by adding the corresponding entries in each vector. Here's an example in R²:

By multiplying a vector v by a real number c you get the scalar multiple cu. It is obtained by multiplying each entry in u by c. For example,
The number c is called a scalar.

The set of vectors containing n entries, where n is a positive integer, is called Rⁿ. Addition and scalar multiplication is performed in the same way for vectors in Rⁿ.

Properties for vectors in Rⁿ:

1. u + v = v + u
2. (u + v) + w = u + (v + w)
3. u + 0 = 0 + u = u
4. u + (-u) = -u + u = 0, where -u = (-1)u
5. c(u + v) = cu + cv
6. (c + d)u = cu + du
7. c(du) = (cd)u
8. 1u = u

To subtract u from v, we write u - v instead of u + (-1)v

The linear combination of vectors v₁, v₂, ..., v_p in Rⁿ with weights c₁, c₂, ..., c_p is given as
The weights can be any real numbers, including zero.