Vectorieel Product
Shanae Maerz
Het kruisproduct, vectorproduct, vectorieel product, uitwendig product of uitproduct, niet te verwarren met het Engelse 'outer product', dat een tensorproduct is, van twee vectoren in drie dimensies is een vector die loodrecht staat op beide vectoren, en waarvan de grootte gelijk is aan het product van de groottes van de beide vectoren en de sinus van de hoek tussen de twee vectoren. De richting van het kruisproduct wordt vastgelegd door de kurkentrekker- of de rechterhandregel. In tegenstelling tot het inwendig product, is het kruisproduct geen scalair, maar een vector.
Regels een en twee houden in dat de richting van het kruisproduct bepaald wordt door de kurkentrekkerregel op de vectoren a \displaystyle \mathbf a en b \displaystyle \mathbf b toe te passen. Tegenwoordig spreekt men ook wel van de rechterhandregel. Regel drie legt de grootte van het kruisproduct vast als gelijk aan de oppervlakte van het parallellogram met de vectoren a \displaystyle \mathbf a en b \displaystyle \mathbf b als zijden.
De determinantformule geeft ook een betekenis aan het kruisproduct in de driedimensionale cordinatenruimte over een willekeurige commutatieve ring R \displaystyle R , dus niet alleen over de rele getallen, en het is op deze manier mogelijk het kruisproduct voor meer dimensies te definiren.
De eerste en de derde eigenschap samen betekenen dat, voor een willekeurig lichaam, in Belgi: veld, K \displaystyle K met willekeurige karakteristiek, de ruimte K 3 \displaystyle K^3 met het kruisproduct een lie-algebra vormt.
Het kruisproduct wordt in de wiskunde vaak gebruikt om met behulp van twee gegeven vectoren een vector te bepalen die loodrecht op de twee eerste staat, onder andere om een normaalvector mee te bepalen.
Het kruisproduct in R 3 \displaystyle \mathbb R ^3 blijft bewaard onder een isometrische lineaire transformatie, 'op het teken na': orintatiebehoudende isometrien, de rotaties, behouden het kruisproduct, orintatie-omkerende isometrien, rotatie-inversies, bijvoorbeeld spiegelingen, veranderen het kruisproduct van twee vectoren in zijn tegengestelde.
The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero length, then their cross product is zero.[2]
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation (or "handedness") of the space (it is why an oriented space is needed). The resultant vector is invariant of rotation of basis. Due to the dependence on handedness, the cross product is said to be a pseudovector.
The cross product a b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule[1] and a magnitude equal to the area of the parallelogram that the vectors span.[2]
In 1842, William Rowan Hamilton first described the algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors.
In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector, William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations.[11] These alternative names are still widely used in the literature.
These equalities, together with the distributivity and linearity of the cross product (though neither follows easily from the definition given above), 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:
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:
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 perpendicularity in the same way that the dot product is a measure of parallelism. 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 dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel.
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).
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:
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
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:
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 where n = 3, combining these two equations results in the expression for the magnitude of the cross product in terms of its components:[15]
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.[18] 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.[18]
From the general properties of the cross product follows immediately that [ a ] a = 0 \displaystyle [\mathbf a ]_\times \,\mathbf a =\mathbf 0 and a T [ a ] = 0 \displaystyle \mathbf a ^\mathrm T \,[\mathbf a ]_\times =\mathbf 0 and from fact that [a] is skew-symmetric it follows that b T [ a ] b = 0. \displaystyle \mathbf b ^\mathrm T \,[\mathbf a ]_\times \,\mathbf b =0.
where the indices i , j , k \displaystyle i,j,k correspond to vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as
In classical mechanics: representing the cross product by using the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are isotropic. (An 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).[citation needed]
When doing this for a y \displaystyle a_y the next two elements down should "wrap around" the matrix so that after the z component comes the x component. For clarity, when performing this operation for a y \displaystyle a_y , the next two components should be z and x (in that order). While for a z \displaystyle a_z the next two components should be taken as x and y.
The 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.
When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. 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 polar vectors, one must take into account that the result is an axial vector. Therefore, for consistency, the other side must also be an axial vector.[citation needed] More generally, the result of a cross product may be either a polar vector or an axial vector, depending on the type of its operands (polar vectors or axial vectors). Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product:
Because the cross product may also be a polar 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 polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a vector triple product involving three polar vectors is a polar vector.
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