向量叉乘(Cross Product)-文档资料
Cross product
The cross-product in respect to a right-handed coordinate systemCross productFrom Wikipedia, the free encyclopediaIn mathematics, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by a × b (where a and b are two given vectors). It results in a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.If the vectors have the same direction or one has zero length, 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 (i.e. a × b = −b × a ) and is distributive over addition (i.e. a × (b + c ) = a × b + a × c ). 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 alsodepends 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.[1]Contents◾1 Definition◾2 Names◾3 Computing the cross product◾3.1 Coordinate notation◾3.2 Matrix notation◾4 Properties◾4.1 Geometric meaning◾4.2 Algebraic properties◾4.3 Differentiation◾4.4 Triple product expansion◾4.5 Alternative formulation◾4.6 Lagrange's identity◾5 Alternative ways to compute the cross product◾5.1 Conversion to matrix multiplication◾5.2 Index notation for tensors◾5.3 Mnemonic◾5.4 Cross visualization◾6 Applications◾6.1 Computational geometryFinding the direction of the crossproduct by the right-hand rule◾6.2 Mechanics◾6.3 Other◾7 Cross product as an exterior product◾8 Cross product and handedness◾9 Generalizations◾9.1 Lie algebra◾9.2 Quaternions◾9.3 Octonions◾9.4 Wedge product◾9.5 Multilinear algebra◾10 History◾11 See also◾12 Notes◾13 References◾14 External linksDefinitionThe cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b . In physics, sometimesthe notation a ∧b is used,[2] though this is avoided in mathematicsto avoid confusion with the exterior product.The cross product a × b is defined as a vector c that is perpendicularto both a and b , with a direction given by the right-hand rule and amagnitude equal to the area of the parallelogram that the vectorsspan.The cross product is defined by the formula[3][4]where θ is the angle between a and b in the plane containing them(hence, it is between 0° and 180°), ǁa ǁ and ǁb ǁ are the magnitudes of vectors a and b , and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule (illustrated). 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.By convention, 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 = −(a × b ). By pointing the forefinger toward 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.The cross product a ×b (vertical, in purple) changes as the angle between the vectors a (blue) and b (red) changes. The cross product is always perpendicular to both vectors, and has magnitude zero when the vectors are parallel and maximum magnitude ǁa ǁǁbǁ when they are perpendicular.According to Sarrus' rule, the determinant of a 3×3 matrix involves multiplications between matrix elements identified by crossed diagonals Using the cross product requires the handedness of the coordinatesystem to be taken into account (as explicit in the definition above).If a left-handed coordinate system is used, the direction of thevector n is given by the left-hand rule and points in the oppositedirection.This, however, creates a problem because transforming from onearbitrary reference system to another (e.g., a mirror imagetransformation from a right-handed to a left-handed coordinatesystem), should not change the direction of n . The problem isclarified by realizing that the cross-product of two vectors is not a(true) vector, but rather a pseudovector . See cross product andhandedness for more detail.NamesIn 1881, JosiahWillard Gibbs, andindependentlyOliver Heaviside,introduced both thedot product and thecross product using a period (a . b ) and an "x" (a x b ),respectively, to denote them.[5]In 1877, to emphasize the fact that the result of a dotproduct is a scalar while the result of a cross product is avector, William Kingdon Clifford coined the alternativenames scalar product and vector product for the two operations.[5] 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 .Conversely, a dot product a · b involves multiplications between corresponding components of a and b . As explained below, the cross product can be expressed in the form of a determinant of a special 3×3 matrix. According to Sarrus' rule, this involves multiplications between matrix elements identified by crossed diagonals.Computing the cross productCoordinate notationThe standard basis vectors i , j , and ksatisfy the following equalities:which imply, by the anticommutativity of the cross product, thatStandard basis vectors (i , j , k , also denoted e 1, e 2,e 3) and vector components of a (a x , a y , a z , alsodenoted a 1, a 2, a 3)The definition of the cross product also implies that(the zerovector).These equalities, together with the distributivity andlinearity of the cross product (but both do not followeasily from the definition given above), are sufficient todetermine the cross product of any two vectors u and v .Each vector can be defined as the sum of threeorthogonal components parallel to the standard basisvectors:Their cross product u × vcan be expanded using distributivity:This can be interpreted as the decomposition of u × v 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 theabove-mentioned equalities and collecting similar terms, we obtain:meaning that the three scalar components of the resulting vector s = s 1i + s 2j + s 3k = u × vareUsing column vectors, we can represent the same result as follows:Figure 1. The area of a parallelogramas a cross productFigure 2. Three vectors defining aparallelepipedMatrix notationThe cross product can also be expressed as the formal [note 1]determinant:This determinant can be computed using Sarrus' rule or cofactor expansion. Using Sarrus' rule, it expandstoUsing cofactor expansion along the first row instead, it expands to[6]which gives the components of the resulting vector directly.PropertiesGeometric meaningThe magnitude of the cross product can be interpreted as thepositive area of the parallelogram having a and b as sides (seeFigure 1):Indeed, one can also compute the volume V of a parallelepipedhaving a , b and c as sides by using a combination of a cross productand a dot product, called scalar triple product (see Figure 2):Since the result of the scalar triple product may be negative, thevolume of the parallelepiped is given by its absolute value. Forinstance,Because the magnitude of the cross product goes by the sine ofthe angle between its arguments, the cross product can bethought of as a measure of ‘perpendicularity’ in the same waythat the dot product is a measure of ‘parallelism’. Given two unitvectors, their cross product has a magnitude of 1 if the two areperpendicular and a magnitude of zero if the two are parallel.The opposite is true for the dot product of two unit vectors.Cross product distributivity over vectoraddition. The vectors b and c are resolvedinto parallel and perpendicular componentsto a : parallel components vanish in thecross product, perpendicular ones remain.The planes indicate the axial vectorsnormal to those planes, and are notbivectors.[7]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◾If the cross product of two vectors is the zero vector, (a ×b = 0), then either one or both is the zero vector, (a = 0and/or b = 0) or else they are parallel or antiparallel (a ||b ) so that the sine of the angle between them is zero (θ =0° or θ = 180° and sin θ = 0).◾The self cross product of a vector is the zero vector, i.e.,a × a = 0.◾The cross product is anticommutative,◾distributive over addition,◾and compatible with scalar multiplication so that ◾It is not associative, but satisfies the Jacobi identity:Distributivity, linearity and Jacobi identity show that the R 3 vector space 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: that is, a × b = a × c with a ≠ 0 does not imply b = c, but only that:From the definition of the cross product, the angle between a and b − c must be zero, and these vectorsmust be parallel. That is, they are related by a scale factor t, leading to:for some scalar t .◾If a · b = a · c and a × b = a × c, for non-zero vector a, then b = c, asandso 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 matrixtransformations:where is a 3 by 3 matrix and is the transpose of the inverse◾The cross product of two vectors lies in the null space of the 2×3 matrix with the vectors as rows:◾For the sum of two cross products, the following identity holds:DifferentiationThe product rule applies to the cross product in a similar manner:This identity can be easily proved using the matrix multiplication representation.Triple product expansionThe cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined asIt 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: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 formulaThe 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, iswhere ∇2 is the vector Laplacian operator.Another identity relates the cross product to the scalar triple product:Alternative formulationThe cross product and the dot product are related by: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:the above given relationship can be rewritten as follows:Invoking the Pythagorean trigonometric identity one obtains: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.[8]Lagrange's identityThe relation:can be compared with another relation involving the right-hand side, namely Lagrange's identity expressed as:[9]where a and b may be n-dimensional vectors. This also shows that the Riemannian volume form for surfaces is exactly the surface element from vector calculus. In the case n=3, combining these two equations results in the expression for the magnitude of the cross product in terms of its components:[10]The same result is found directly using the components of the cross-product found from:In R3 Lagrange's equation is a special case of the multiplicativity |vw| = |v||w| of the norm in the quaternion algebra.It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the Binet-Cauchy identity:[11][12]If a = c and b = d this simplifies to the formula above.Alternative ways to compute the cross productConversion to matrix multiplicationThe vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:[11]where superscript T refers to the transpose operation, and [a]× is defined by:Also, if a is itself a cross product:thenProof by substitutionEvaluation of the cross product givesHence, the left hand side equalsNow, for the right hand side,And its transpose isEvaluation of the right hand side givesComparison shows that the left hand side equals the right hand side.This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors.This notation is also often much easier to work with, for example, in epipolar geometry.From the general properties of the cross product follows immediately thatandand from fact that [a]× is skew-symmetric it follows thatThe above-mentioned triple product expansion (bac-cab rule) can be easily proven using this notation.The above definition of [a ]× means that there is a one-to-one mapping between the set of 3×3 skew-symmetric matrices, also known as the Lie algebra of SO(3), and the operation of taking the cross product with some vector a .In , denoting with, the canonical base vectors, the cross product of a generic vector withis given by:, where These matrices share the following properties:◾(skew-symmetric);◾Both trace and determinant are zero;◾;◾ (see below);The orthogonal projection matrix of a vector is given by . The projection matrix onto the orthogonal complement is given by, where is the identity matrix. For thespecial case of, it can be verified thatwhere (slight change in notation). For other properties of orthogonal projection matrices, see projection (linear algebra).Index notation for tensorsThe cross product can alternatively be defined in terms of the Levi-Civita symbol εijk and a dot product ηmi (= δmifor an orthonormal basis), which are useful in converting vector notation for tensor applications:where the indices correspond to vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention asSpecial casesin which repeated indices are summed over the values 1 to 3. Note that this representation is another form of the skew-symmetric representation of the cross product:In classical mechanics: representing the cross-product with the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are isotropic. (Quick example: consider a particle in a Hooke's Law potential in three-space, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the cross-product-represented angular momentum, which are made clear by the abovementioned Levi-Civita representation).MnemonicThe word "xyzzy" can be used to remember the definition of the cross product.Ifwhere:then:The second and third equations can be obtained from the first by simply vertically rotating the subscripts, x → y → z → x. The problem, of course, is how to remember the first equation, and two options are available for this purpose: either to remember the relevant two diagonals of Sarrus's scheme (those containing i), or to remember the xyzzy sequence.Since the first diagonal in Sarrus's scheme is just the main diagonal of the above-mentioned matrix, the first three letters of the word xyzzy can be very easily remembered.Cross visualizationSimilarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may help you to remember the correct cross product formula.Ifthen:If we want to obtain the formula for we simply drop the and from the formula, and take the nexttwo components down -It should be noted that when doing this for the next two elements down should "wrap around" the matrixso that after the z component comes the x component. For clarity, when performing this operation for ,the next two components should be z and x (in that order). While for the next two components should be taken as x and y.For then, if we visualize the cross operator as pointing from an element on the left to an element on theright, we can take the first element on the left and simply multiply by the element that the cross points to in the right hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as well. This results in our formula -We can do this in the same way for and to construct their associated formulas. ApplicationsComputational geometryThe cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points , and . It corresponds to the direction of the cross product of the two coplanar vectors defined by the pairs of points and , i.e., by the sign of theexpression . In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation aroundfrom to , otherwise a negative angle. From another point of view, the sign of tells whether lies to the left or to the right of line .MechanicsMoment of a force applied at point B around point A is given as:OtherThe cross product in relation to the exterior product. In red are the orthogonal unit vector, and the "parallel" unit bivector.The cross product occurs in the formula for the vector operator curl. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product.The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints.Cross product as an exterior productThe cross product can be viewed in terms of the exterior product.This view allows for a natural geometric interpretation of the crossproduct. In exterior algebra the exterior product (or wedge product)of two vectors is a bivector. A bivector is an oriented plane element,in much the same way that a vector is an oriented line element.Given two vectors a and b , one can view the bivector a ∧b as theoriented parallelogram spanned by a and b . The cross product isthen obtained by taking the Hodge dual of the bivector a ∧b ,mapping 2-vectors to vectors:This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented line element – a vector – whereas, for example, in 4 dimensions the Hodge dual of a bivector is two-dimensional –another oriented plane element. So, only in three dimensions is thecross product of a and b the vector dual to the bivector a ∧b : it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as a ∧b has relative to the unit bivector; precisely the properties described above.Cross product and handednessWhen 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 anarbitrary 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.GeneralizationsThere are several ways to generalize the cross product to the higher dimensions.Lie algebraThe cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity. Many Lie algebras exist, and their study is a major field of mathematics, called Lie theory. For example, the Heisenberg algebra gives another Lie algebra structure on In the basis theproduct isQuaternionsThe cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on R3. The unit vectors i, j, k correspond to "binary" (180 deg) rotations about their respective axes (Altmann, S. L., 1986, Ch. 12), said rotations being represented by "pure" quaternions (zero scalar part) with unit norms.For instance, the above given cross product relations among i, j, and k agree with the multiplicative relations among the quaternions i, j, and k. In general, if a vector [a1, a2, a3] is represented as the quaternion a1i + a2j + a3k, the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.Alternatively, using the above identification of the 'purely imaginary' quaternions with R3, the cross product may be thought of as half of the commutator of two quaternions.OctonionsA cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.Wedge productIn general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the wedge product in three dimensions by using the Hodge dual to map 2-vectors to vectors. The Hodge dual of the wedge product yields an (n−2)-vector, which is a natural generalization of the cross product in any number of dimensions.The wedge product and dot product can be combined (through summation) to form the geometric product.Multilinear algebraIn the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form,[note 2] a (0,3)-tensor, by raising an index.In detail, the 3-dimensional volume form defines a product by taking the determinant of thematrix given by these 3 vectors. By duality, this is equivalent to a function (fixing any two inputs gives a function by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism and thus this yields a map which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index". Translating the above algebra into geometry, the function "volume of the parallelepiped defined by" (where the first two vectors are fixed and the last is an input), which defines a function , can be represented uniquely as the dot product with a vector: this vector is the cross product From this perspective, the cross product is defined by the scalar triple product,In the same way, in higher dimensions one may define generalized cross products by raising indices of the n-dimensional volume form, which is a -tensor. The most direct generalizations of the cross product are to define either:◾a -tensor, which takes as input vectors, and gives as output 1 vector – an -aryvector-valued product, or◾a -tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rankn−2 – a binary product with rank n−2 tensor values. One can also define -tensors for other k. These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity.The -ary product can be described as follows: given vectors in define theirgeneralized cross product as:◾perpendicular to the hyperplane defined by the◾magnitude is the volume of the parallelotope defined by the which can be computed as the Gram determinant of the◾oriented so that is positively oriented.This is the unique multilinear, alternating product which evaluates to , and so forth for cyclic permutations of indices.In coordinates, one can give a formula for this -ary analogue of the cross product in R n by:。
向量的计算知识点与公式总结
向量的计算知识点与公式总结，文本结构清晰向量是一种物理方面的重要概念，它在各种科学问题的研究中被广泛使用，其中最重要的是向量计算，它是对两个向量进行算术运算，例如加减乘除、求模和叉乘，以表示两个向量在空间中的特性和相互关系。
向量的计算主要分为点积 (Dot product) 与叉乘 (Cross Product) 两个计算方式。
点积是两个向量的数量乘积，叉乘是两个向量的向量积，两者的计算公式分别为：点积： $A \cdot B =|A||B|cos(\theta) =AxBx+AyBy+AzBz$ (其中$A = (Ax,Ay,Az)$和 $B = (Bx,By,Bz)$ ）叉乘：$A \times B =(AyBz-AzBy, AzBx-AxBz, AxBy-AyBx)$从点积的计算公式可以看出，点积是一个数量乘积，它等于两个向量平行时的数量积，它可以表明两个向量的夹角。
叉乘的计算公式可以看出，它是两个向量的向量积，它可以衡量两者之间的相对方位，可以求出两个向量的法向量，从而确定法线方程。
此外，还有两个重要的运算方式——欧几里德距离 (Euclidean Distance)和曼哈顿距离 (Manhattan Distance)。
欧几里德距离是最短距离的定义，计算公式如下：$D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$（其中$A =(x_1,y_1,z_1)$和 $B = (x_2,y_2,z_2)$ ），用于表示两点之间的距离，它可以用来衡量两点之间的相对距离。
曼哈顿距离是一种比较简单的距离度量，计算公式如下：$D = |x_2-x_1|+|y_2-y_1|+|z_2-z_1|$（其中$A = (x_1,y_1,z_1)$和 $B = (x_2,y_2,z_2)$），它可以表示两点之间的绝对距离，可以用来比较两点之间的距离大小。
总的来说，向量的计算是研究向量在空间中的几何特征和相互关系的重要工具，有点积、叉乘、欧几里德距离和曼哈顿距离等多种运算方式。
向量-向量叉乘向量点乘
向量-向量叉乘向量点乘2010年07月28日星期三14:33向量（Vector）在几乎所有的几何问题中，向量（有时也称矢量）是一个基本点。
向量的定义包含方向和一个数（长度）。
在二维空间中，一个向量可以用一对x和y来表示。
例如由点（1,3）到（5,1的向量可以用（4,-2）来表示。
这里大家要特别注意，我这样说并不代表向量定义了起点和终点。
向量仅仅定义方向和长度。
向量加法向量也支持各种数学运算。
最简单的就是加法。
我们可以对两个向量相加，得到的仍然是一个向量。
我们有：V1 （x1, y1）+V2 （x2, y2）=V3（x1+x2, y1+y2）下图表示了四个向量相加。
注意就像普通的加法一样，相加的次序对结果没有影响（满足交换律），减法也是一样的。
The sum of vectors A+B+C+D点乘（Dot Product）如果说加法是凭直觉就可以知道的，另外还有一些运算就不是那么明显的，比如点乘和叉乘。
点乘比较简单，是相应元素的乘积的和：V1（ x1, y1）V2（x2, y2） = x1*x2 + y1*y2注意结果不是一个向量，而是一个标量（Scalar）。
点乘有什么用呢，我们有：A B = |A||B|Cos（0）B是向量A和向量B见的夹角。
这里|A|我们称为向量A的模（norm）,也就是A的长度，在二维空间中就是|A| = sqrt（x2+y2）。
这样我们就和容易计算两条线的夹角：Cos（0 ） = A B /（|A||B|）当然你知道要用一下反余弦函数acos（）啦。
（回忆一下cos（90）=0和cos（0） = 1还是有好处的，希望你没有忘记。
）这可以告诉我们如果点乘的结果，简称点积，为0的话就表示这两个向量垂直。
当两向量平行时，点积有最大值另外，点乘运算不仅限于2维空间，他可以推广到任意维空间。
（译注：不少人对量子力学中的高维空间无法理解，其实如果你不要试图在视觉上想象高维空间，而仅仅把它看成三维空间在数学上的推广，那么就好理解了）叉乘(cross product)相对于点乘，叉乘可能更有用吧。
向量叉乘的运算公式
向量叉乘的运算公式向量的叉乘是在向量运算中的一种重要运算，也被称为向量积或外积。
它在三维空间中的向量叉乘是基于向量的线性性质和右手法则来定义的。
在本文中，我们将讨论向量的叉乘运算公式以及它的性质和应用。
向量的叉乘是一个叉积向量，它的结果是一个新的向量，垂直于参与运算的两个向量，并且其大小等于两个参与向量所构成平行四边形的面积。
向量的叉乘可以用符号“×”表示，如A×B。
向量的叉乘可以用叉乘运算公式进行计算。
假设我们有两个向量A和B，它们的坐标表示为A=(a₁,a₂,a₃)和B=(b₁,b₂,b₃)。
则它们的叉乘结果向量C=(c₁,c₂,c₃)的计算公式如下：c₁=a₂b₃-a₃b₂c₂=a₃b₁-a₁b₃c₃=a₁b₂-a₂b₁这个公式的推导是基于向量的线性性质和几何直观的原则。
具体来说，c₁是通过a₂b₃和a₃b₂的差来计算的，而c₂和c₃分别是通过类似的方式计算得到的。
这个公式的结果是一个新的向量C，其方向垂直于A和B所在的平面，并遵循右手法则。
这个公式可以进一步化简为矩阵形式。
我们可以将向量A和B分别表示为列矩阵A=[a₁,a₂,a₃]ᵀ和B=[b₁,b₂,b₃]ᵀ，其中ᵀ表示矩阵的转置。
那么向量的叉乘公式可以写成如下的矩阵形式：C=[a]×[B]=[ijk][a₁a₂a₃][b₁b₂b₃]其中[i,j,k]是单位向量，分别代表x、y和z轴的正方向。
这个矩阵形式的公式是非常方便和紧凑的，特别适合在计算机程序中实现。
向量的叉乘具有以下的性质：1.叉乘是一个满足反对称性的运算，即A×B=-(B×A)。
这个性质意味着叉乘的结果与向量的顺序有关。
2.叉乘满足分配律，即(A+B)×C=A×C+B×C。
这个性质表明叉乘对于向量的加法是分配的。
3. 叉乘结果向量的大小等于两个参与向量构成平行四边形的面积，即，A×B， = ，A，B，sin(θ)，其中，A，和，B，分别表示向量A和B 的大小，θ表示A和B之间的夹角。
向量的叉乘运算法则
向量的叉乘运算法则向量的叉乘是向量运算中的一种重要方式，它在物理学、工程学、计算机图形学等领域都有着广泛的应用。
在本文中，我们将介绍向量的叉乘运算法则，包括定义、性质和计算方法。
1. 定义。
给定两个三维向量a和b，它们分别可以表示为：a = (a1, a2, a3)。
b = (b1, b2, b3)。
那么a和b的叉乘结果记为a×b，它是一个新的向量，其分量由以下公式计算得出：a×b = (a2b3 a3b2, a3b1 a1b3, a1b2 a2b1)。
2. 性质。
向量的叉乘具有以下性质：（1）反交换律，a×b = -b×a。
（2）分配律，a×(b+c) = a×b + a×c。
（3）数量积与叉乘的关系，a×b与a·b垂直，且|a×b| = |a|·|b|·sinθ，其中θ为a和b之间的夹角。
3. 计算方法。
为了计算向量的叉乘，可以使用行列式的方法。
给定向量a和b，它们的叉乘结果可以表示为：a×b = |i j k|。
|a1 a2 a3|。
|b1 b2 b3|。
其中|i j k|表示行列式，它分别对应x、y、z轴的单位向量。
通过展开行列式，可以得到a×b的分量表达式。
另外，也可以直接使用分量的公式进行计算，即a×b = (a2b3 a3b2, a3b1 a1b3, a1b2 a2b1)。
4. 应用。
向量的叉乘在物理学和工程学中有着广泛的应用。
例如，在力学中，叉乘可以用来计算力矩；在电磁学中，叉乘可以表示磁感应强度；在计算机图形学中，叉乘可以用来计算法向量等。
总之，向量的叉乘是一种重要的向量运算方式，它具有明确的定义和性质，可以通过行列式或分量公式进行计算，并且在多个领域都有着重要的应用。
通过深入理解向量的叉乘运算法则，我们可以更好地应用它解决实际问题，推动科学技术的发展。
点乘与叉乘
点乘目录点乘叉乘点乘点乘（dot product），也叫向量的内积、数量积。
顾名思义，求下来的结果是一个数。
向量a·向量b=|a||b|cos<a,b> （ <a,b> 指向量a与向量b之间的夹角）在物理学中，已知力与位移求功，实际上就是求向量F与向量s的内积，即要用点乘。
坐标化表示将向量用坐标表示（三维向量），若向量a=(a1,b1,c1)，向量b=(a2,b2,c2)，则，向量a·向量b=a1a2+b1b2+c1c2运用（一）点乘可用于判断向量垂直判断条件：在向量a与向量b的模皆不为0的情况下，向量a·向量b=0由向量a·向量b=|a||b|cos<a,b>可很容易的得出当|a| 、|b|皆不为0时，cos<a,b>为0也即向量a与向量b互相垂直。
（二）关于用点乘判断向量平行的误区判断平行：向量a·向量b=|a|*|b|；而非向量a·向量b=1（×）由向量a·向量b=|a||b|cos<a,b>可很容易的得出叉乘叉乘（cross product），也叫向量的外积、向量积。
顾名思义，求下来的结果是一个向量，记这个向量为c。
|向量c|=|向量a×向量b|=|a||b|sin<a,b> （ <a,b> 指向量a与向量b之间的夹角）向量c的方向与a,b所在的平面垂直，且方向要用“右手法则”判断（用右手的食指先表示向量a的方向，然后中指朝着手心的方向摆动到向量b的方向，大拇指所指的方向就是向量c的方向）。
因此向量的外积不遵守乘法交换率，因为向量a×向量b= - 向量b×向量a在物理学中，已知力与力臂求力矩，就是向量的外积，即叉乘。
坐标化表示将向量用坐标表示（三维向量），若向量a=(a1,b1,c1)，向量b=(a2,b2,c2)，则，向量a×向量b= | i j k ||a1 b1 c1||a2 b2 c2|=(b1c2-b2c1,c1a2-a1c2,a1b2-a2b1)（i、j、k分别为空间中相互垂直的三条坐标轴的单位向量）。
向量叉积公式
向量叉积公式
公式：a × b = |a| * |b| * sinθ 叉乘又叫向量的外积、向量积。
点乘和叉乘的区别：
点乘，也叫向量的内积、数量积。
顾名思义，求下来的结果是一个数。
向量a · 向量b=|a||b|cos。
在物理学中，已知力与位移求功，实际上就是求向量F与向量s的内积，即要用点乘。
叉乘，也叫向量的外积、向量积。
顾名思义，求下来的结果是一个向量，记这个向量为c。
|向量c|=|向量a×向量b|=|a||b|sin。
向量c的方向与a，b所在的平面垂直，且方向要用“右手法则”判断(用右手的四指先表示向量a的方向，然后手指朝着手心的方向摆动到向量b的方向，大拇指所指的方向就是向量c的方向)。
向量的外积不遵守乘法交换率，因为向量a×向量b=-向量b×向量a。
向量的三种乘法
向量的三种乘法
向量的三种乘法包括点乘（也称为内积或数量积）、叉乘（也称为向量积或外积）和外展（也称为广义的叉积）。
以下是这三种乘法的详细介绍：
点乘（Dot Product）：也叫向量的内积、数量积。
两个n维向量a和b的点积定义为：a·b = a1b1+a2b2+...+anbn。
点乘的几何意义是一个向量在另外一个向量上的投影。
点乘的结果是一个标量，表示两个向量的相似度，两个向量越“相似”，它们的点乘越大。
叉乘（Cross Product）：也叫向量积，数学中又称外积、叉积，物理中称矢积。
叉乘是两个三维向量之间的运算，其结果是一个向量，模长等于两个向量模长的乘积与它们之间夹角正弦值的乘积，方向垂直于这两个向量所在的平面，且遵守右手定则。
外展（Outer Product）：对于任意n维向量a和b，外展的结果是一个n×n的矩阵，其元素aij为ai和bj的乘积。
总的来说，点乘主要用来衡量两个向量的相似度，叉乘主要用来生成一个与已有两个向量都垂直的新向量，而外展则可以将一个向量转化为一个矩阵，这在一些数学和物理计算中非常有用。