In mathematics, a vector is an element of a vector space, and as such the central object of study of linear algebra. This is a general definition including a variety of mathematical objects (numbers, sequences, functions, and operators). In physics and engineering the term vector usually refers to the particular vectors from Euclidean space (or from tangent spaces of a differentiable manifold), and represent quantities characterized by both magnitude and direction. Examples are displacement, velocity, momentum, force and acceleration. One also consideres bound or fixed vectors which are characterized by magnitude, direction and base point. Examples of these are position, torque and angular momentum. These two types of vectors are discussed below. Vector fields can be thought off as rules that assign a bound vector to every point in space. Examples are the velocity field of a moving fluid or the magnetic or electrical field.
Graphical representation and vector operations
Vectors are typically graphically represented as arrows: the arrowhead points into the direction of the vector and the length of the arrow represents the vector's magnitude.
Symbols standing for vectors are usually printed in boldface as a; this is also the convention adopted in this encyclopedia. Other conventions write an arrow above or a line beneath the letter. Alternatively, vectors - especially those dealing with distances or force diagrams - can be written as AB with an arrow above or a line beneath; here A denotes the base point and B denotes the tip of the arrow.
Two vectors a and b may be added together graphically by placing the 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. This addition method is sometimes called the parallelogram rule. 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).
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. 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 operations deserves the name "subtraction" because (a - b) + b = a.
A vector may also be multiplied by a real number. Numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. Multiplying a vector by a scalar "scales" the length of the vector by the amount of the scalar. If the scalar is negative, it also changes the direction of the vector by 180o. The result of multiplying the vector a by the scalar r is denoted by ra. 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.
Coordinate systems and coordinates
In order to calculate with vectors, the graphical representation is too cumbersome. One introduces a coordinate system of two (if dealing with plane vectors) or three (if dealing with vectors in familiar three-dimensional space) mutually perpendicular unit vectors. A unit vector is a vector of magnitude 1. If the unit vectors are called i, j and k, then any vector a 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:
/ a1 \ a = | a2 | or a = [a1 a2 a3] \ a3 /
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.
Coordinate systems facilitate the vector operations considerably: adding or subtracting two vectors only involves adding or subtracting their coordinates and multiplying a vector by a real number involves multiplying its coordinates with that same number.
The length or magnitude of the vector a is denoted by |a|. In arrow representation, length is easy to find since this representation is based on direction and magnitude. If a coordinate system has been introduced, the length of the vector a = a1i + a2j + a3k can be computed as
- |a| = √(a12 + a22 + a32)
which is a consequence of the Pythagorean Theorem.
The dot product of two vectors a and b, also called the scalar product since its result is a scalar, is denoted by a·b or sometimes by (a, b) and is defined as:
- a·b = |a||b| cos(θ)
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 definition has the following consequences:
- the dot product is symmetric, i.e. a·b = b·a.
- two non-zero vectors a and b are perpendicular if and only if a·b = 0
- the dot product is bilinear, i.e. a·(rb + c) = r (a·b) + (a·c)
From these it follows directly that the dot product of two vectors a = [a1 a2 a3] and b = [b1 b2 b3] given in coordinates can be computed particularly easily:
- a·b = a1b1 + a2b2 + a3b3
or, using matrix multiplication and treating the vectors as 1-by-3 matrices:
- 'a·b = a'bT
where bT denotes the transpose of the matrix b.
The dot product satisfies all the axioms of an inner product.
The cross product or vector 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 and seven dimensions. In three dimensions the cross product, denoted a×b, is a vector perpendicular to both a and b and is defined as:
- a×b = |a||b| sin(θ) n
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. 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 pseudo-vector. 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.
General properties of the cross product include:
- the cross product is anti-symmetric, which means: a×b = -b×a
- the cross product is not associative, but satisfies the Jacobi identity:
- a×(b×c) + b×(c×a) + c×(a×b) = 0
- a×(b×c) = (a·c)b - (a·b)c
- the cross product is distributive across addition, meaning that a×(b + c) = a×b + a×c
- the cross product is compatible with scalar multiplication in the following sense: (ra)×b = a×(rb) = r(a×b).
- two non-zero vectors a and b are parallel if and only if a×b = 0.
- i×j = k, j×k = i, k×i = j
- It follows that in Cartesian coordinates, the cross product can be written as:
- a×b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1]
- the above component notation can also be written as the determinant of the matrix: i j k a1 a2 a3 b1 b2 b3
In seven dimensions it is also possible to define a cross product. This product has the following properties in common with the usual 3-dimensional cross product:
- It is bilinear in the sense that x×(ay+bz)=ax×y+bx×z and (ay+bz)×x=ay×x+bz×x
- It is anti-commutative: x×y=-y×x
and it satisfies the Jacobi identity above. Unfortunately the definition is more complex than that in three dimensions. The 3-D and 7-D cross products are related to the quaternions and octonions, respectively.
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:
- (a b c) = a·(b×c)
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 off 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:
- (a b c) = (c a b) = (b c a) = -(a c b) = -(b a c) = -(c b a)