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## Vector (spatial)
The concept of a Often informally described as an object with a "magnitude" (size) and "direction", a vector is more formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below.
Such a vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a
## DefinitionsInformally, avector is a quantity, characterized by a number (indicating size or "magnitude") and a direction, that is often represented graphically by an arrow. Examples are "moving north at 90 m.p.h" or "pulling towards the center of Earth with a force of 70 Newtons".
The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration. Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar.
A related concept is that of a pseudovector (or
Sometimes, one speaks informally of ## GeneralizationsIn mathematics, avector over a field k is any element of a vector space. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of R^{d} in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!## Representation of a vector
Symbols standing for vectors are usually printed in boldface as length or magnitude or norm of the vector a is denoted by |a|.Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction.
In the figure above, the arrow can also be written as or
In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a
a_{1}, a_{2} and a_{3} on the specific choice of coordinate system i, j and k.
the length of the vector
## Vector Equality## Vector Addition and Subtraction
Let b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below:
This addition method is sometimes called the
The difference of
b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a - b, as illustrated below:
If
A vector may also be multiplied by a real number
ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180^{o}. Two examples (r = -1 and r = 2) are given below:
Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated. ## Dot Product
The
a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a. This operation is often useful in physics; for instance, work is the dot product of force and displacement.## Cross Product
The cross product (also
a and b, and n is a unit vector perpendicular to both a and b. The problem with this definition is that there are two unit vectors perpendicular to both b and a. Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the coordinate system. The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by
In such a system,
The length of ## Scalar Triple ProductThescalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as:
a, b and c are oriented like the coordinate system i, j and k.In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense:
pseudoscalar: under a coordinate inversion (x goes to -x), it flips sign.## External links- Online vector identities (pdf)
- Vectors at Wikibooks
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