Euclidean vector

This article is about the concept of vector in mathematics. For the concept in biology, see biological vector.

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Informally, a vector is a quantity, characterized by a number (indicating size or "magnitude") and a direction, that is often represented by an arrow. Examples are "moving north at 90 m.p.h" or "pulling towards the center of Earth with 70 Newtons".

In differential geometry, physics and engineering, the term vector usually refers to the particular geometric vectors from Euclidean space (or from tangent spaces of a differentiable manifold), quantities characterized by both magnitude and direction. Examples are displacement, velocity, momentum, force, torque and acceleration. Under this definition, a vector is a special type of tensor—tensor of contravariant rank one. (In this sense, the concept of "direction" is formalized by saying that the vector components transform like the coordinates under rotations.)

(In linear algebra, a vector is any element of a vector space. This is a general definition including a variety of mathematical objects (numbers, sequences, functions, and operators). Note that under this definition, a tensor is a special vector! )

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.

One also considers bound or fixed vectors which are characterized by magnitude, direction and base point. Examples of these are position (commonly called position vector), torque and angular momentum.

Vector fields can be thought of as rules that assign a bound vector to every point in space. Examples are the velocity field of a moving fluid or the electric field.

A related concept is that of a pseudo-vector (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 pseudo-vectors include the magnetic field and the angular momentum. (This distinction is usually overlooked, however, although it becomes important in studying symmetry properties.)

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 ${\vec {a}}$ or a . 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", the "base", the "start", or the "origin"; and the point B is called the "head", the "tip", the "endpoint", or the "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 ${\vec {AB}}$ or AB

In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean spaces can be represented as a linear combination of n mutually prependicular 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:

$\mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\\end{pmatrix}}\quad {\mbox{ or }}\quad \mathbf {a} ={\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\end{pmatrix}}$

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.

the length of the vector a = a₁i + a₂j + a₃k can be computed as

$\left|\mathbf {a} \right|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}$

which is a consequence of the Pythagorean Theorem.

Vector Equality

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:

$\mathbf {a} +\mathbf {b} =(a_{1}+b_{1})\mathbf {i} +(a_{2}+b_{2})\mathbf {j} +(a_{3}+b_{3})\mathbf {k}$

The addition may be represented 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, 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:

$\mathbf {a} -\mathbf {b} =(a_{1}-b_{1})\mathbf {i} +(a_{2}-b_{2})\mathbf {j} +(a_{3}-b_{3})\mathbf {k}$

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:

File:Vector Subtraction.png

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 r. Numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is:

$r\mathbf {a} =(ra_{1})\mathbf {i} +(ra_{2})\mathbf {j} +(ra_{3})\mathbf {k}$

The length of 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:

File:Vector Scalar Multiplication.png

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.

Dot Product

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:

$\mathbf {a} \cdot \mathbf {b} =\left|\mathbf {a} \right|\left|\mathbf {b} \right|\cos(\theta )$

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.

Cross Product

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:

$\mathbf {a} \times \mathbf {b} =\left|\mathbf {a} \right|\left|\mathbf {b} \right|\sin(\theta )\mathbf {n}$

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. Graphically the cross product can be represented by

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:

$(\mathbf {a} \ \mathbf {b} \ \mathbf {c} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )$

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 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:

$(\mathbf {a} \ \mathbf {b} \ \mathbf {c} )=(\mathbf {c} \ \mathbf {a} \ \mathbf {b} )=(\mathbf {b} \ \mathbf {c} \ \mathbf {a} )=-(\mathbf {a} \ \mathbf {c} \ \mathbf {b} )=-(\mathbf {b} \ \mathbf {a} \ \mathbf {c} )=-(\mathbf {c} \ \mathbf {b} \ \mathbf {a} )$

External Link

Online vector identities (pdf)

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Vector is also the name of a 1980s new wave rock band. See Vector (band)