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VECTOR ANALYSIS (Basic Theoretical Background) 1.1 VECTOR OPERATIONS Hereafter, vector quantities will be written in bold, while scalar quantities will be written in italic. 1.1.1 Vector operations The basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field, and consist of: scalar multiplication multiplication of a scalar field, a,=and a vector field, v, yielding a vector field: a=bv =;= vector addition = addition of two vector fields, yielding a vector field: = 12 =b v +v =;= dot product = multiplication of two vector fields, yielding a scalar field: 12 b= ⋅vv (or 12 b= ×vv ) ;= (the symbol ×=is alternative to the symbol ⋅)= cross product = multiplication=of two vector fields, yielding a vector field:== 12 = ×bv v =(or = 12 = ∧bv v )=;= (the symbol ∧=is alternative to the symbol ×)= There are also two=triple products := scalar triple product = the dot product of a vector and a cross product of two vectors: = 1 123 () b =⋅×vvv (or = 11 23 () b =×∧v vv )=;= vector triple product = the cross product of a vector and a cross product of two vectors:== 1 1 23 () =×× b v vv or 2 3 21 () =×× b v vv == Or=alternatively: 11 23 () =∧∧ bv vv or 2 3 21 () =∧∧ b v vv =;= although these are less often used as basic operations, as they can be expressed in terms of the dot and cross= products.= 1.2 SCALAR MULTIPLICATION Not to be confused with scalar product. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term “scalar” itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and must be distinguished from inner product of two vectors (where the product is a scalar). 1.2.1 Definition In general, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is denoted cv .= … omissis 1.2.3 Properties Scalar multiplication obeys the following rules (vector in boldface): ()cd c d+v= v + v = • Left distributivity: ()cd c d+ v= v + v =;= • Right distributivity: ()c cc v+w = v + w =;= • Associativity: () ( )cd c d v= v =;= • Multiplying by 1 does not change a vector: 1v= v ;= • Multiplying by 0 gives the null vector: 0v= 0 ;= • Multiplying by −1 gives the additive inverse: ( 1)−−v= v .= = Here + is=addition=either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the=multiplication=operation in the field.= 1.3 VECTOR ADDITION In mathematics, physics and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or – as here – simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors. It is important to distinguish Euclidean vectors from the more general concept in linear algebra of vectors as elements of a vector space. General vectors in this sense are fixed-size, ordered collections of items as in the case of Euclidean vectors, but the individual items may not be real numbers, and the normal Euclidean concepts of length, distance and angle may not be applicable. (A vector space with a definition of these concepts is called an inner product space). In turn, both of these definitions of vector should be distinguished from the statistical concept of a random vector. The individual items in a random vector are individual real- valued random variables, and are often manipulated using the same sort of mathematical vector and matrix operations that apply to the other types of vectors, but otherwise usually behave more like collections of individual values. Concepts of length, distance and angle do not normally apply to these vectors, either; rather, what links the values together is the potential correlations among them. 1.3.1 Vector representations Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a. (Uppercase letters are typically used to represent matrices). Other conventions include a=or= a, especially in handwriting.= Alternatively, some use a=tilde=(~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as or AB. Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. Here the point A is called the origin, tail, base, or initial point; point B is called the head, tip, endpoint, terminal point or final point. The length of the arrow is proportional to the vector’s magnitude, while the direction in which the arrow points indicates the vector’s direction. ⊗ ,  On a two-dimensional diagram, sometimes a vector perpendicular to a plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299  ) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it=(Unicode U+2297 ⊗ ) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an=arro∂=head on and viewing the vanes of an arrow from the back.= = = A vector in the Cartesian plane, showing the position =of=a=point A with coordinates (2,3) In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n-dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system. The endpoint of a vector can be identified with an ordered list of n real numbers (n - tuple). These numbers are the coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system, and are typically called the scalar components (or scalar projections) of the vector on the axes of the coordinate system. As an example in two dimensions (see figure), the vector from the origin O = (0,0) to the point A = (2,3) is simply written as: = (2, 3)a = The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation= =is usually not deemed necessary and very rarely used.= In three dimensional Euclidean space (or= 3 ), vectors are identified with triples of scalar components:= 12 3 = (a ,a ,a ) a = also written=as:= xy≠ = (a ,a ,a ) a = These numbers are often arranged into a=column vector or row vector, particularly when dealing with= matrices, as follows:= 1 2 3 a = a a     a = = [ ] 12 3 =a a aa = Another way to represent a vector in=n-dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them: 1= ( 1, 0, 0 ) e , 2= ( 0,1, 0 ) e , 3= ( 0, 0,1 ) e These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis of a Cartesian coordinate system, respectively. In terms of these, any vector a in 3 can be expressed in the form 12 3 123 = ( a , a , a ) = a (1, 0, 0) + a (0,1, 0) + a (0, 0,1) a or 1 2 3 11 2 2 3 3 = ( + + ) = a +a +aa aa a e e e where a 1, a 2, a 3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x, y, and z (see figure), while a 1, a 2, a 3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often instead denoted i, j, k , (or �x, �y, �z , in which the hat symbol ^ typically denotes unit vectors). In this case, the scalar and vector components are denoted respectively xa , ya , za , and xa , ya , za (note the difference in boldface). Thus, xyz x y z = ( + + ) = a +a +aa aaa i j k The notation ie is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. 1.3.2 Decomposition As explained above a vector is often described by a set of vector components that are mutually perpendicular and add up to form the given vector. Typically, these components are the projections of the vector on a set of reference axes (or basis vectors). The vector is said to be decomposed or resolved with respect to that set. Illustration of tangential and normal components of a vector to a surface. However, the decomposition of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as ˆx, � y, �z as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of the unit vectors of a cylindrical coordinate system � � � ρφ( , ,z) or spherical coordinate system �� � θφ (r, , ) . The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively. The choice of a coordinate system doesn't affect the properties of a vector or its behaviour under transformations. A vector can be also decomposed with respect to "non-fixed" axes which change their orientation as a function of time or space. For example, a vector in three dimensional space can be decomposed with respect to two axes, respectively normal, and tangent to a surface (see figure). Moreover, the radial and tangential components of a vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it. In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame). 1.3.3 Basic properties The following section uses the Cartesian coordinate system with basis vectors: 1= ( 1, 0, 0 ) e =, 2= ( 0,1, 0 ) e =, 3= ( 0, 0,1 ) e = and assumes that all vectors have the origin as a common base point. A vector a will be written as: 11 2 2 3 3 = a +a +a ae e e = 1.3.3.1 Equality Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors: 11 2 2 3 3 = a +a +aae e e =or = 12 3 = a +a +a ai j k = an≤= 11 2 2 3 3 = b +b +b be e e =or = 12 3 b= b +b +bi jk = are equal if:= 11a = b , 22a = b , = 33a = b 1.3.3.2 Addition and subtraction Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is: 11 2 2 3 3 + = (a + b ) + (a + b ) + (a + b ) ab i j k = 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 that have the same base point, it will 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: 11 2 2 3 3 = (a b ) + (a b ) + (a b ) −− − −ab i j k = Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the end points 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: 1.3.3.3 Scalar multiplication A vector may also be multiplied, or re-scaled, by a real number r. In the context of conventional vector algebra, these real numbers are often called scalars (from scale) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called scalar multiplication. The resulting vector is: 12 3 = (a)+(a) +(a) rr r rai jk = Intuitively, multiplying by a scalar=r stretches a vector out by a factor of r. Geometrically, this can be visualized (at least in the case when r is an integer) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector. If r is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (r = −1 and r = 2) are given below: Scalar multiplication of a vector by a factor of 3 stretches the vector out The scalar multiplications 2 a and − a of a vector a Scalar multiplication is distributive over 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. Length The length or magnitude or norm of the vector a is denoted by ||a|| or, less commonly, |a|, which is not to be confused with the absolute value (a scalar "norm"). The length of the vector a can be computed with the Euclidean norm: 2221 23 = aaa ++ a = which=is a consequence of the=Pythagorean theorem=since the basis vectors e 1, e 2, e 3 are orthogonal unit vectors. This happens to be equal to the square root of the dot product, discussed below, of the vector with itself: = ⋅ a aa = Unit vector Main article: Unit vector A unit vector is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as normalizing a vector. A unit vector is often indicated with a hat as in ˆa.= To normalize a vector [ ] 12 3 =a a aa , scale the vector by the reciprocal= of its length a . That is:= 12 3aa a �==++ a a i jk aa a a = = The normalization of a= vector=a into a unit vector ˆa Null vector Main article: Null vector The null vector (or zero vector) is the vector with length zero. Written out in coordinates, the vector is (0,0,0), and it is commonly denoted 0  , or= 0, or simply 0. Unlike any other vector it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector which is a multiple of the null vector). The sum of the null vector with any vector=a is a (that is, 0 + a = a ). 1.4 DOT PRODUCTS Reference: http://en.wikipedia.org/wiki/Dot_product 1.4.1 Definition The dot product is often defined in one of two ways: algebraically or geometrically. Equivalence of these definitions is proven later. 1.4.1.1 Algebraic definition The dot product of two vectors a = [a 1, a 2, ..., a n] and b = [b 1, b 2, ..., b n] is defined as: i i 11 2 2 1 = n nn i ab ab a b a b = ⋅ = + ++ ∑ ab  = where ∑ denotes summation notation=and n is the dimension of the vector space. • In two-dimensional space, the dot product of vectors [ a, b ] and [ c, d ] is ac + bd. Similarly, in a three dimensional space, the dot product of vectors [ a, b, c ] and [ d, e, f ] is ad + be + cf. Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix. 1.4.1.2 Geometric definition In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by A . The dot product of two Euclidean vectors= A and B is defined by: = cos θ ⋅⋅ AB A B = where θ is the angle between A and B (with 0 180 θ °≤ ≤ ° ).= In particular, if A and B are orthogonal, then the angle between them is 90°, so in that case: =0 ⋅ AB = At the other extreme, if=B = A, then the angle is 0°, and the dot product is just the length of A squared: 2 = ⋅ AA A = 1.4.2 Scalar projection and the equivalence of the definitions The scalar projection of a Euclidean vector A onto a Euclidean vector B is given by: = cos θ BAA where θ is the angle between A and B. In terms of the geometric definition of the dot product, this can be rewritten: ˆ = ⋅ BA AB where ˆ=BB B is the unit vector in the direction of B. Scalar projection The dot product is thus characterized geometrically by: == ⋅ BA AB A B B A The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α, ( ) = ( )= ( )αα α ⋅ ⋅⋅ A B AB A B It also satisfies a distributive law, meaning that: ( + )= +⋅ ⋅⋅ A B C AB AC Distributive law for the dot product As a consequence, if 1,, n ee  are the standard basis vectors in n , then writing: [ ] 1 = ,, A n ii i =∑ AA A e  [ ] 1 = ,, B n ii i =∑ BB B e  we have: B( ) B Ai i ii ii ⋅= ⋅ = ∑∑ AB Ae which is precisely the algebraic definition of the dot product. More generally, the same identity holds with the e i replaced by any orthonormal basis. 1.4.3 Properties The dot product fulfills the following properties if a, b, and c are real vectors and r is a scalar. 1) Commutative: ⋅=⋅ab ba 2) Distributive over a vector addition: ( +) + ⋅ =⋅⋅ a bc abac 3) Bilinear: ( + ) ( )+( )rr ⋅ =⋅⋅a b c ab ac 4) Scalar multiplication: 1 2 12 ( ) ( ) ( )( )c c cc ⋅= ⋅ a b ab Properties 3 and 4 follow from 1 and 2. 5) Orthogonal: Two non-zero vectors a and b are orthogonal if and only if 0 ⋅= ab.= 6)=No cancellation:= Unlike multiplication of ordinary numbers,= where if=ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law: If ⋅=⋅ab ac =and 0≠a , then we can write:= ⋅a (b − c) = 0 by the distributive law, the result above says this just means that a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠ c. 7) Invariance under isometric changes of basis: Provided that the basis is orthonormal, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed. The above mentioned geometric interpretation relies on this property. In other words, for an orthonorma space with any number of dimensions, the dot product is invariant under a coordinate transformation based on an orthogonal matrix. This corresponds to the following two conditions: o The new basis is again orthonormal (i.e., it is orthonormal expressed in the old one). o The new base vectors have the same length as the old ones (i.e., unit length in terms of the old basis). If a and b are functions, then the derivatives of ⋅ab =is= ⋅⋅ a' b + a b' = 1.4.4 Application to the cosine law Main article: law of cosines Given two vectors a and b separated by angle θ (see image right), they form a triangle with a third side c = a - b. The dot product of this with itself is: 222 2 (- )(- )= - - + = a - - +b a -2 +b⋅= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = ⋅ cc a b a b aa ab ba bb ab ab ab = 2 22c a + b - 2 (a b) cos θ = = which is the law of cosines.= = Triangle with vector edges=a and b, separated by angle θ. 1.4.5 Triple product expansion Main article: Triple product This is a very useful identity (also known as Lagrange's formula) involving the dot- and cross-products. It is written as: ( ) ( )- ( ) × ×= ⋅ ⋅ a b c bac c ab ===or = ( ) ( )- ( ) ∧ ∧= × ×a b c ba c ca b = which is easier to remember as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula is commonly used to simplify vector calculations in physics.= = 1.5 VECTOR OPERATIONS Refernce: http://en.wikipedia.org/wiki/Cross_product In mathematics, the cross product, vector product, or Gibbs’ vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering. If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular for perpendicular vectors this is a rectangle and the magnitude of the product is the product of their lengths. The cross product is anticommutative, distributive over addition and satisfies the Jacobi identity. The space and product form an algebra over a field, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on the choice of orientation or “handedness”. The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. = The cross-product in respect to a right-handed coordinate system= = 1.5.1 Definition The cross product of two vectors a and b is denoted by × ab . In physics, sometimes the notation ∧ab =is used, though is= avoided in mathematics to avoid confusion with the=exterior product. The cross product ×ab =is defined as a vector=c that is perpendicular to both a and b, with a direction given by the right-hand-rule and a magnitude equal to the area of the parallelogram that the vectors span. The cross product is defined by the formula: sin θ ×= ab a b n where θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), ‖a‖ and ‖b‖ are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the d irection given by the right -hand rule as illustrated. Finding the direction of the cross product by the right-hand-rule If the vectors a and b are parallel (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0. The direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb (see the picture on the right). Using this rule implies that the cross-product is anti-commutative, i.e., =- ( ) ××b a ab . By pointing the forefinger towar≤=b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector. Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction. This, however, creates a problem because transforming from one arbitrary reference system to another (e.g., a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of n. The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. See cross-product and handedness for more detail. 1.5.2 Names The cross product is also called the vector product or Gibbs' vector product. The name Gibbs' vector product is after Josiah Willard Gibbs, who around 1881 introduced both the dot product and the cross product, using a dot (a · b) and a cross (a × b) to denote them. To emphasize the fact that the result of a dot product is a scalar, while the result of a cross product is a vector, Gibbs also introduced the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature. Both the cross notation (a × b) and the name cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. According to Sarrus’ rule, the determinant of a 3×3 matrix involves multiplications between matrix elements identified by crossed diagonals. Conversely, a dot product a · b involves multiplications between corresponding components of a and b. As explained below, the cross product can be defined as the determinant of a special 3×3 matrix. According to Sarrus’ rule, this involves multiplications between matrix elements identified by crossed diagonals. 1.5.3 Computing the cross product 1.5.3.1 Coordinate notation The standard basis vectors i, j, and k satisfy the following equalities: ×=ijk ×=jk i ×= ki j which imply, by the anticommutativity of the cross product, that: ×=−ji k ×=− kj i ×=−ik j Standard basis vectors (i, j, k, also denoted e 1, e2, e3) and vector components of a (a x, ay, az, also denoted a 1, a2, a3) The definition of the cross product also implies that ×=× = × =ii jj kk 0 (the zero vector) These equalities, together with the distributivity and linearity of the cross product, are sufficient to determine the cross product of any two vectors a and b. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors: 1 2 31 2 3 aa a =+ += + +aa a a i j k 1 2 31 2 3 bb b =+ += + + bb b b i j k = Their cross product a × b can be expanded using distributivity: 12 3 1 2 3 111 21 3 212 22 3 313 23 3 (a a a ) (b b b ) ab ab ab ab ab ab ab ab ab ×= + + × + + = ×+ ×+ × + ×+ ×+ × + ×+ ×+ × ab i j k i j k ii i j ik ji j j jk ki k j kk = This can be interpreted as the decomposition of a × b into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above mentioned equalities and collecting similar terms, we obtain: 11 1 2 1 3 212 2 2 3 31 3 23 3 2 3 3 2 31 1 3 1 2 21 ab ab ab( ) ab( ) ab ab ab ab ( ) ab (a b a b ) (a b a b ) (a b a b ) × = + + −+ −+ + + + −+ = − +− +− ab 0 k j k 0i j i0 ijk = meaning that the three=scalar components=of the resulting vector c = c 1i + c 2j + c 3k = a × b are 1 23 32c ab ab = − = 2 31 1 3c a b ab = − = 3 12 21c ab a b = − = Using=column vectors, we can represent the same result as follows:= 1 23 32 2 31 1 3 3 12 21 c ab ab c a b ab c ab a b −     = −    −   = Matrix notation The definition of the cross product can also be represented by the determinant of a formal matrix: 12 3 1 23 aa a bbb ×= i jk ab = This determinant can be computed using=Sarrus� rule=or=cofactor expansion.= Using Sarrus∇=rule, it expands to:= 2 3 31 1 2 3 2 1 3 21a b a b ab a b ab a b ×= + + − − −ab i j k i j k = Using cofactor expansion along the first row instead, it expands to:= 2 3 13 12 2 3 13 12 a a aa aa b b bb bb ×= − −ab i j k = which gives the components of the resulting vector directly.= Properties Geometric meaning See also: Triple product = Figure 1: = The area of a parallelogram as a cross product = = Figure 2: = Three vectors defining a parallelepipe≤ = = The=magnitude=of the cross product can be interpreted as the positive=area=of the=parallelogram=having=a and b as sides (see Figure 1): sin A θ =×= ab a b = Indeed, one can also compute the volume V of a parallelepiped having a, b and c as sides by using a combination of a cross product and a dot product, called scalar triple product (see Figure 2): ( ) ( ) ( ) ⋅ × =⋅× =⋅×a bc b ca c ab = Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its= absolute value. For instance,= () V=⋅×a bc = Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure=of "perpendicularness" in the same way that the=dot product=is a= measure of "parallelness". Given two=unit vectors, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The opposite is true for the dot product of two= unit vectors.= Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product= of the two unit vectors yields the sine (which will always be positive).= Algebraic properties The cross product is anticommutative, × =−×ab ba = distributive=over addition,= ( ) ( ) ( ) ++ × =××a bc ab ac = and compatible with scalar multiplication so that:= ( ) ( ) ( ) r rr ×=× = × a b a b ab = It is not=associative=but satisfies the=Jacobi identity:= ( ) ( ) ( ) × × + × × +× × =a bc b ca c ab 0 = Distributivity, linearity and Jacobi identity show that 3 together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3). The cross product does not obey the cancellation law: a × b = a × c with non-zero a does not imply that b = c. Instead if a × b = a × c: ( ) ( ) ( ) = ×− ×=× −0 ab ac a b c If neither a nor b - c is zero then from the definition of the cross product the angle between them must be zero and they must be parallel. They are related by a scale factor, so one of b or c can be expressed in terms of the other, for example: () t = + c ba for some scalar t. If a · b = a · c and a × b = a × c, for non-zero vector a, then b = c, as: ()× −= a bc 0 () 0 ⋅ −= a bc so b − c is both parallel and perpendicular to the non-zero vector a, something that is only possible if b − c = 0 so they are identical. From the geometrical definition the cross product is invariant under rotations about the axis defined by a × b. More generally the cross product obeys the following identity under matrix transformations: ( ) ( ) ( ) ( ) det T− ×= × a bab M M MM where M is a 3 by 3 matrix and T − M is the transpose of the inverse. The cross product of two vectors in 3-D always lies in the null space of the matrix with the vectors as rows ( ) NS  ×∈  a ab b For the sum of two cross products, the following identity holds: ( ) ( ) +++ ××=−×− ××abcd a c b d ad cb Differentiation Main article: Vector-valued function: Derivative and vector multiplication The product rule applies to the cross product in a similar manner: ( ) + dd d dx dx dx ×= × × ab ab ba This identity can be easily proved using the matrix multiplication representation. Triple product expansion Main article: Triple product The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as: ( ) ⋅×abc It is the signed volume of the parallelepiped with edges a, b and c and as such the vectors can be used in any order that’s an even permutation of the above ordering. The following therefore are equal: ()()()⋅ × =⋅× =⋅× abc bca cab The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula: ()()()×× = ⋅− ⋅ a b c b a c c ab = The mnemonic="BAC minus CAB" is used to remember the order of the vectors in the right hand member.= This formula is used in physics to simplify vector calculations. A special case, regarding=gradients=and useful= in=vector calculus, is:= ( ) ( ) ( ) ( ) 2 ∇× ∇× = ∇ ∇⋅ − ∇⋅∇ = ∇ ∇⋅ −∇ f f f ff = ∂here= 2∇ is the vector Laplacian operator.= = = The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence ( ) ∇⋅ =of the gradient ( ) ∇ f . Thus, if=f is a twice-differentiable real-valued function, then the Laplacian of f is defined by: ( ) 2 ∆ = ∇ = ∇⋅∇ ff f = where the latter notations derive from formally writing:= 1nxx  ∂∂ ∇= + + ∂∂  = Equivalently, the Laplacian of=f is the sum of all the unmixed second partial derivatives in the Cartesian coordinates x i: 2 2 1 n i ix = ∂ ∆= ∂ ∑ f f = = The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence ( ) ∇⋅ =of the gra≤ient= ( ) ∇ f . Thus, if=f is a twice- differentiable real-valued function, then the Laplacian of f is defined by: () 2 ∆ = ∇ = ∇⋅∇ff f = Another identity relates the cross product to the scalar triple product:= ( ) ( ) ( ) ( ) × ×× =⋅×ab ac aac a = Alternative formulation The cross product and the dot product are related by: ( ) 2 22 2 × = −⋅a b a b ab = The right-hand side is the=Gram determinant=of=a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle θ between the two vectors, as: cos θ ⋅= ab a b = the above given relationship can be rewritten as follows:= ( ) 2 22 2 1 cos θ ×=− −ab a b = Invoking the Pythagorean trigonometric identity one obtains: sin θ ×=−a b ab = which is the magnitude=of the cross product expressed in terms of=θ, equal to the area of the parallelogram defined by a and b (see definition above). The combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product Cross product and handedness When measurable quantities involve cross products, the handedness of the coordinate systems used cannot be arbitrary. However, when physics laws are written as equations, it should be possible to make an arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two vectors, one must take into account that when the handedness of the coordinate system is not fixed a priori, the result is not a (true) vector but a pseudovector. Therefore, for consistency, the other side must also be a pseudovector. More generally, the result of a cross product may be either a vector or a pseudovector, depending on the type of its operands (vectors or pseudovectors). Namely, vectors and pseudovectors are interrelated in the following ways under application of the cross product: • vector × vector = pseudovector • pseudovector × pseudovector = pseudovector • vector × pseudovector = vector • pseudovector × vector = vector. So by the above relationships, the unit basis vectors i, j and k of an orthonormal, right-handed (Cartesian) coordinate frame must all be pseudovectors (if a basis of mixed vector types is disallowed, as it normally is) since i × j = k, j × k = i and k × i = j. Because the cross product may also be a (true) vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a (true) vector and the other one is a pseudovector (e.g., the cross product of two vectors). For instance, a vector triple product involving three (true) vectors is a (true) vector. A handedness-free approach is possible using exterior algebra. In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. Geometrically, it is the opposite, of equal magnitude but in the opposite direction (versus), of its mirror image. This is as opposed to a true or polar vector, which on reflection matches its mirror image. In three dimensions the vector c is associated with the cross product of two polar vectors a and b: = ×cab = The vector=c calculated this way is a pseudovector. One example is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors a and b, which can be said to span the plane. The vector a × b is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which) and is a pseudovector. Notes In linear algebra, geometry, and physics, the term versor is often used for a right versor. In this case, a versor is defined as a unit vector indicating the orientation of a directed axis in a Cartesian coordinate. Reference: http://en.wikipedia.org/wiki/Vectorial_analysis APPENDIX Basic theory Vector a (or a) Vector b (or b) Versors (or unit vectors ) i and j Versors (or unit vectors ) i , j and k Components (or projections) of a 2 -D vector a Components (or projections) of a 3-D vector a Vector addition Vector additi on = = = = Vector addition = Vector addition = = ( ) ( ) x yxx xxc c ab ab = + =+= + + +c i jab i j = = = = Vector= difference = Vector= difference = = = = = Tw o-dimensional space vectors (2D) and complex notation Tw o-dimensional space vector (2D) Dot product: a · b (or a × b) Cros s product: a × b (or ∧ ab ) = = = = = = Moment of a force with respect to pole O = = = = = = Tangential velocity of a point mass moving along a circular trajectory = = = =