In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a "magnitude" and a "direction". The word vector is also now used for more general concepts (see also vector and generalizations below), but in this article we describe the original spatial meaning except where otherwise noted.
A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law.
A vector can be 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.
A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus.
Informally, a vector is a quantity characterized by a number (indicating magnitude) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/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 R, so that a coordinate vector x is transformed to x' = Rx, then any other vector v is similarly transformed via v' = Rv.
More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.)
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 axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector.
Sometimes, one speaks informally of bound or fixed vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities.
In mathematics, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of Rd 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 a; this is also the convention adopted in this encyclopedia. Other conventions includes or a, especially in handwriting. The 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:
Here the point 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 AB
In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. In this article, we will consider R3 as an example. In R3, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R3 can be written as a = a1i + a2j + a3k with real numbers a1, a2 and a3 which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix:
even though this notation suppresses the dependence of the coordinates a1, a2 and a3 on the specific choice of coordinate system i, j and k.
Length of a vector
The length of the vector a = a1i + a2j + a3k can be computed as
which is a consequence of the Pythagorean theorem.
Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about bound vector, then two bound vectors are equal if they have the same base point and end point.
For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (unbounded) vector.
Vector addition and subtraction
Let a=a1i + a2j + a3k and b=b1i + b2j + b3k.
The sum of a and b is:
The addition may be represented graphically by placing the start of the arrow 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 parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c).
The difference of a and b is:
Subtraction of two vectors can be geometrically defined as follows: to subtract 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 and b are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a - b) + b = a.
A vector may also be multiplied by a real number r. Numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is:
The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180o. 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: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b.
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.
A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use it to create a unit vector. This is known as normalizing a vector.
To normalize a vector, scale the vector by the inverse of its length. That is:
The dot product of two vectors a and b (also called the inner product, or, since its result is a scalar, the scalar product) is denoted by a·b or sometimes by (a, b) and is defined as:
where θ is the measure of the angle between 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.
The cross product (also vector product or outer product) differs from the dot product primarily in that the result of a cross product of two vectors is a vector.
While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions (although a related product exists in seven dimensions - see below).
The cross product, denoted a×b, is a vector perpendicular to both a and b and is defined as:
where θ is the measure of the angle between 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 this figure
In such a system, a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product.
The length of a×b can be interpreted as the area of the parallelogram having a and b as sides.
Scalar triple product
The scalar 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:
It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors 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:
Technically, the scalar triple product isn't a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign.