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A vector space is a set over a field that has to satisfy certain rules, called axioms. The field in question can be Z2 (see discussion), but unlike a field, a vector's inverse is distinct from the vector. Therefore, in order to satisfy the "inverse elements of addition" axiom for vector spaces, a vector space must minimally (except if it is the null space) have three vectors, v, 0, and v-1. The null space only has one vector, 0.
Field's can allow for two distinct elements, unlike vector spaces, because for any given element of a field, for example a, a + (-a) = 0 meets the inverse axiom, but a and -a aren't required to be distinct. They are simply scalar magnitudes, unlike vectors which can often be thought of as having a direction attached to them. That's why the vectors, v and -v are distinct, because they're pointing in opposite directions.
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What is an under vector room?
There does not seem to be an under vector room, but there is vector space. Vector space is a structure that is formed by a collection of vectors. This is a term in mathematics.
How do you do cross products?
You need to know that the cross product of two vectors is a vector perpendicular to both vectors. It is defined only in 3 space. The formula to find the cross product of vector a (vector a=[a1,a2,a3]) and vector b (vector b=[b1,b2,b3]) is: vector a x vector b = [a2b3-a3b2,a3b1-a1b3,a1b2-a2b1]
Why null vector is called vector?
The same sort of reasoning that zero is a number. It ensures that the set of all vectors is closed under addition and that, in turn, allows the generalization of many operations on vectors.Also, the way we got around the concept of having something with zero magnitude also have a direction is pretty cool. We made it up! In abstract algebra it's perfectly OK to constrain a specific algebraic structure with rules (called axioms) that the structure must follow.In your example, the algebraic structure that vectors are in is called a "vector space." One of the axioms that define a vector space is:"An element, 0, called the null vector, exists in a vector space, v, such that v + 0 = vfor all of the vectors in the vector space."Ta Da!! Aren't we clever?
What is cross products?
in mathematics the cross products are the binary operation on two vectors in a 3dimensional Euclidean space that results in another vector which is perpendicular to the containing the 2 inputs vector.
What is the difference between zero and vector zero?
The zero vector occurs in any dimensional space and acts as the vector additive identity element. It in one dimensional space it can be <0>, and in two dimensional space it would be<0,0>, and in n- dimensional space it would be <0,0,0,0,0,....n of these> The number 0 is a scalar. It is the additive identity for scalars. The zero vector has length zero. Scalars don't really have length. ( they can represent length of course, such as the norm of a vector) We can look at the distance from the origin, but then aren't we thinking of them as vectors? So the zero vector, even <0>, tells us something about direction since it is a vector and the zero scalar does not. Now I think and example will help. Add the vectors <2,2> and <-2,-2> and you have the zero vector. That is because we are adding two vectors of the same magnitude that point in opposite direction. The zero vector and be considered to point in any direction. So in summary we have to state the obvious, the zero vector is a vector and the number zero is a scalar.
