Vector (spatial)
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- This article discusses vectors that have a particular relation to the spatial coordinates. For a generalization, see vector space. For other uses, see vector.
In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector has properties that do not depend on the coordinate system used to describe it. However, a vector is often described by a fixed number of components, each of which is dependent upon the particular coordinate system being used, such as Cartesian coordinates, spherical coordinates or polar coordinates.
A common example of a vector is force — it has a magnitude and an orientation and multiple forces sum according to the parallelogram law.
A spatial 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.
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Definitions
Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) 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. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below).
Accordingly, let, for example, each of two vectors be expressed as three space coordinates, and apply the formula for the cross product, resulting in three coordinates, which represent a third vector. If we rewrite the two vectors in rotated coordinates, and apply the formula for the cross product again, then the result is the original cross product in terms of rotated coordinates.
Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field. If we rewrite the original vector field in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values, and apply the formula for the curl based on these functions, then the result is the rewritten version of the original curl: also in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values.
The same applies for dot product, gradient, divergence, vector addition and scalar multiplication. For these, also reflection in a plane can be applied. The scalars involved should not be transformed (e.g. in the case of a rotation by 180°, the scalar should not be multiplied by -1). Thus even in 1D we have to distinguish scalars and vectors: 2 × 3 = 6 can be interpreted as a scalar multiplication or a dot product, but not as a product of two vectors. Similarly differentiation in 1D can be interpreted as a gradient or a divergence: one of the two functions is scalar and one a vector, and the argument is a vector, ensuring invariance under inversion of the vectors without changing the scalars.
Since rotation of the three Cartesian coordinate axes changes the formulas the same as an inverse rotation of the field itself, we can also conclude:
- if the same rotation is applied to two vectors, then the cross product is correspondingly rotated, but the dot product remains the same
- rotation of a scalar field results in a correspondingly rotated vector field for the gradient
- rotation of a vector field results in a correspondingly rotated scalar field for the divergence and a correspondingly rotated vector field for the curl
where rotation of a scalar field involves only rotation of the position vectors, while rotation of a vector field involves also a corresponding rotation of the vector field values. Note that the concept of corresponding rotations applies even if different coordinate systems are used for field values and position vectors, so that e.g. for one we multiply by an orthogonal matrix and for the other we add an angle to an angle coordinate.
In order to use the usual formulas, e.g. to compute mechanical work, the x-axis of forces should be in the same direction as the x-axis of position, etc. When, as described above, coordinate rotations of position are accompanied by corresponding coordinate rotations of forces, this property is preserved. On the other hand, the origin of forces is simply at the zero force (no force), while the origin of position can be chosen as desired. For example, work depends on displacement, which is the difference of positions and therefore does not depend on the origin.
Position and function value of a vector field are often, but not necessarily, expressed in similar coordinate systems. For example gravitational field strength due to a particular point mass may be <math>g_r = -9.8 (r/r_0)^{-2} \ \mbox{m}/\mbox{s}^2</math>, with both the function value and the position vector in spherical coordinates. For the position vector the origin is chosen here at the center of the point mass; for the field strength the origin is simply at "zero field strength" anyway. How the other two coordinates are chosen does not matter in this case, because the field does not depend on them, and the field has no components in their directions.
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. See also parity (physics).
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.
Examples in one dimension
A force may be "15 N to the right", with coordinate 15 N if the basis vector is to the right, and −15 N if the basis vector is to the left. The magnitude of the vector is 15 N in both cases. A displacement may be "4 m to the right", with coordinate 4 m if the basis vector is to the right, and −4 m if the basis vector is to the left. The magnitude of the vector is 4 m in both cases. The work done by the force in the case of this displacement is 60 J in both cases.
The force and displacement are vectors, the magnitudes are scalars, and the coordinates are neither.
Generalizations
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
Vectors are usually denoted in boldface, as a. Other conventions include <math>\vec{a}</math> or a, especially in handwriting. Alternately, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type.
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 <math>\overrightarrow{AB}</math> or AB.
On a two-dimensional diagram, sometimes a vector perpendicular to plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre indicates a vector pointing out of the front of the diagram, towards the viewer. A circle with a cross inscribed in it indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip an arrow front on and viewing the vanes of an arrow from the back.
The length or magnitude or norm of the vector a is denoted by |a|.
If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1:250 and 1 m:50 N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.
In order to calculate with vectors, the graphical representation may be 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:
- <math>{a} = \begin{bmatrix}
a_1\\ a_2\\ a_3\\
\end{bmatrix} </math>
- <math> {a} = \begin{pmatrix}
a_1 & a_2 & a_3 \\
\end{pmatrix} </math>
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 in a Euclidean space can be computed with the Euclidian norm
- <math>\left\|\mathbf{a}\right\|=\sqrt{a_1^2+a_2^2+a_3^2}</math>
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:
- <math>\mathbf{a}+\mathbf{b}
=(a_1+b_1)\mathbf{i} +(a_2+b_2)\mathbf{j} +(a_3+b_3)\mathbf{k}</math>
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:
- <math>\mathbf{a}-\mathbf{b}
=(a_1-b_1)\mathbf{i} +(a_2-b_2)\mathbf{j} +(a_3-b_3)\mathbf{k}</math>
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.
In physics, vectors of different physical dimension may occur in the same diagram. However, adding or subtracting them (graphically or otherwise) is meaningless.
Scalar multiplication
A vector may also be multiplied by a real number r. In mathematics numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is:
- <math>r\mathbf{a}=(ra_1)\mathbf{i}
+(ra_2)\mathbf{j} +(ra_3)\mathbf{k}</math>
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:
Image:Scalar multiplication of vectors.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.
In physics, scalars also have a unit. The scale of acceleration in the diagram is e.g. 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.
Unit vector
Main article: Unit vector
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.
Image:Vector normalization.png
To normalize a vector a = [a1, a2, a3], scale the vector by the reciprocal of its length ||a||. That is:
- <math>\mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{\hat{i}} + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{\hat{j}} + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{\hat{k}}</math>
Dot product
Main article: Dot product
The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as:
- <math>\mathbf{a}\cdot\mathbf{b}
=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos(\theta)</math>
where ||a|| and ||b|| denote the norm (or length) of a and b, and 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:
- <math>\mathbf{a}\times\mathbf{b}
=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\sin(\theta)\,\mathbf{n}</math>
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:
- <math>(\mathbf{a}\ \mathbf{b}\ \mathbf{c})
=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})</math>
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:
<math>(\mathbf{a}\ \mathbf{b}\ \mathbf{c})</math> <math>=(\mathbf{c}\ \mathbf{a}\ \mathbf{b})</math> <math>=(\mathbf{b}\ \mathbf{c}\ \mathbf{a})</math> <math>=-(\mathbf{a}\ \mathbf{c}\ \mathbf{b})</math> <math>=-(\mathbf{b}\ \mathbf{a}\ \mathbf{c})</math> <math>=-(\mathbf{c}\ \mathbf{b}\ \mathbf{a})</math>
Technically, the scalar triple product is not a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign.
Vectors as directional derivatives
A vector may also be defined as a directional derivative: consider a function <math>f(x^\alpha)</math> and a curve <math>x^\alpha (\sigma)</math>. Then the directional derivative of <math>f</math> is a scalar defined as
<math>\frac{df}{d\sigma} = \frac{dx^\alpha}{d\sigma}\frac{\partial f}{\partial x^\alpha}.</math>
where the index <math>\alpha</math> is summed over the appropriate number of dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to <math>x^\alpha (\sigma)</math>:
<math>t^\alpha = \frac{dx^\alpha}{d\sigma}.</math>
We can rewrite the directional derivative in differential form (without a given function <math>f</math>) as
<math>\frac{d}{d\sigma} = t^\alpha\frac{\partial}{\partial x^\alpha}.</math>
Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. We can therefore define a vector precisely:
- <math>\mathbf{a} \equiv a^\alpha \frac{\partial}{\partial x^\alpha}.</math>
See also
- Four-vector, the specialization to space-time in relativity
- Pseudovector
- Vector calculus
- Vector bundle
- Unit vector
- Null vector
- Normal vector
External links
- Vectors—short practical introduction to vectors at the high school level.
- Online vector identities (PDF)
- Vectors at Wikibooksbg:Вектор
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