Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing directed line segments. The bivector has an attitude (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, it has a magnitude equal to the area of the parallelogram with edges and ), and has an orientation being the side of on which lies within the plane spanned by and . In layman terms, any surface defines the same bivector if it is parallel to the same plane (same attitude), has the same area, and same orientation (see figure).…

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing directed line segments. The bivector has an attitude (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, it has a magnitude equal to the area of the parallelogram with edges and ), and has an orientation being the side of on which lies within the plane spanned by and . In layman terms, any surface defines the same bivector if it is parallel to the same plane (same attitude), has the same area, and same orientation (see figure). Bivectors are generated by the exterior product on vectors: given two vectors and , their exterior product is a bivector, as is any sum of bivectors. Not all bivectors can be expressed as an exterior product without such summation. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product of two vectors is alternating, so is the zero bivector, and , producing the opposite orientation. Concepts directly related to bivectors are rank-2 antisymmetric tensor and skew-symmetric matrix. Associativity : (\mathbf{ab})\mathbf{c} = \mathbf{a}(\mathbf{bc}) Left and right distributivity : \begin{align} \mathbf{a}(\mathbf{b} + \mathbf{c}) &= \mathbf{ab} + \mathbf{ac} \\ (\mathbf{b} + \mathbf{c})\mathbf{a} &= \mathbf{ba} + \mathbf{ca} \end{align} Scalar square : , where is the quadratic form, which need not be positive-definite. \tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}) = \tfrac{1}{2} \left((\mathbf{a} + \mathbf{b})^2 - \mathbf{a}^2 - \mathbf{b}^2\right) is a sum of scalars and so a scalar. From the law of cosines on the triangle formed by the vectors its value is , where is the angle between the vectors. It is therefore identical to the scalar product between two vectors, and is written the same way, \mathbf{a} \cdot \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}). It is symmetric, scalar-valued, and can be used to determine the angle between two vectors: in particular if and are orthogonal the product is zero. \mathbf{a} \wedge \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} - \mathbf{ba}) It is antisymmetric in and \mathbf{b} \wedge \mathbf{a} = \tfrac{1}{2}(\mathbf{ba} - \mathbf{ab}) = -\tfrac{1}{2}(\mathbf{ab} - \mathbf{ba}) = -\mathbf{a} \wedge \mathbf{b} and by addition: \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \wedge \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}) + \tfrac{1}{2}(\mathbf{ab} - \mathbf{ba}) = \mathbf{ab} That is, the geometric product is the sum of the symmetric scalar product and alternating exterior product. To examine the nature of , consider the formula (\mathbf{a} \cdot \mathbf{b})^2 - (\mathbf{a} \wedge \mathbf{b})^2 = \mathbf{a}^2\mathbf{b}^2, which using the Pythagorean trigonometric identity gives the value of (\mathbf{a} \wedge \mathbf{b})^2 = (\mathbf{a} \cdot \mathbf{b})^2 - \mathbf{a}^2\mathbf{b}^2 = \left|\mathbf{a}\right|^2\left|\mathbf{b}\right|^2( \cos^2 \theta - 1) = -\left|\mathbf{a}\right|^2\left|\mathbf{b}\right|^2\sin^2 \theta With a negative square, it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector. It has magnitude , where is the angle between the vectors, and so is zero for parallel vectors. To distinguish them from vectors, bivectors are written here with bold capitals, for example: \mathbf{A} = \mathbf{a} \wedge \mathbf{b} = -\mathbf{b} \wedge \mathbf{a} \,, although other conventions are used, in particular as vectors and bivectors are both elements of the geometric algebra. \mathbf{B} = \mathbf{e}_1 \wedge \mathbf{e}_2 + \mathbf{e}_3 \wedge \mathbf{e}_4 = \mathbf{e}_1\mathbf{e}_2 + \mathbf{e}_3\mathbf{e}_4 = \mathbf{e}_{12} + \mathbf{e}_{34} cannot be written as the exterior product of two vectors. A bivector that can be written as the exterior product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions; in four dimensions every bivector is the sum of at most two exterior products. A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by a directed plane area. \mathbf{A}\mathbf{B} = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B} + \mathbf{A} \wedge \mathbf{B}. The quantity is the scalar-valued scalar product, while is the grade 4 exterior product that arises in four or more dimensions. The quantity is the bivector-valued commutator product, given by \mathbf{A} \times \mathbf{B} = \tfrac{1}{2}(\mathbf{AB} - \mathbf{BA}), The space of bivectors is a Lie algebra over , with the commutator product as the Lie bracket. The full geometric product of bivectors generates the even subalgebra. Of particular interest is the product of a bivector with itself. As the commutator product is antisymmetric the product simplifies to \mathbf{A}\mathbf{A} = \mathbf{A} \cdot \mathbf{A} + \mathbf{A} \wedge \mathbf{A}. If the bivector is simple the last term is zero and the product is the scalar-valued , which can be used as a check for simplicity. In particular the exterior product of bivectors only exists in four or more dimensions, so all bivectors in two and three dimensions are simple. M_B = \begin{pmatrix} 0 & B_{12} & -B_{31} \\ -B_{12} & 0 & B_{23}\\ B_{31} & -B_{23} & 0 \end{pmatrix}. This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector minus the exterior product; an example is the angular velocity tensor. Skew symmetric matrices generate orthogonal matrices with determinant through the exponential map. In particular, applying the exponential map to a bivector that is associated with a rotation yields a rotation matrix. The rotation matrix given by the skew-symmetric matrix above is M_R = \exp{M_B}. The rotation described by is the same as that described by the rotor given by R = \exp{\tfrac{1}{2} B}, and the matrix can be also calculated directly from rotor . In three dimensions, this is given by M_R = \begin{pmatrix} (R\mathbf{e}_1R^{-1}) \cdot \mathbf{e}_1 & (R\mathbf{e}_2R^{-1}) \cdot \mathbf{e}_1 & (R\mathbf{e}_3R^{-1}) \cdot \mathbf{e}_1 \\ (R\mathbf{e}_1R^{-1}) \cdot \mathbf{e}_2 & (R\mathbf{e}_2R^{-1}) \cdot \mathbf{e}_2 & (R\mathbf{e}_3R^{-1}) \cdot \mathbf{e}_2 \\ (R\mathbf{e}_1R^{-1}) \cdot \mathbf{e}_3 & (R\mathbf{e}_2R^{-1}) \cdot \mathbf{e}_3 & (R\mathbf{e}_3R^{-1}) \cdot \mathbf{e}_3 \end{pmatrix}. Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix the eigenvalues can be calculated by solving the characteristic equation for that matrix . By the fundamental theorem of algebra this has three roots (only one of which is real as there is only one eigenvector, i.e., the axis of rotation). The other roots must be a complex conjugate pair. They have unit magnitude so purely imaginary logarithms, equal to the magnitude of the bivector associated with the rotation, which is also the angle of rotation. The eigenvectors associated with the complex eigenvalues are in the plane of the bivector, so the exterior product of two non-parallel eigenvectors results in the bivector (or a multiple thereof). \begin{align} \mathbf{a}\mathbf{b} &= (a_1\mathbf{e}_1 + a_2\mathbf{e}_2)(b_1\mathbf{e}_1 + b_2\mathbf{e}_2) \\&= a_1b_1\mathbf{e}_1\mathbf{e}_1 + a_1b_2\mathbf{e}_1\mathbf{e}_2 + a_2b_1\mathbf{e}_2\mathbf{e}_1 + a_2b_2\mathbf{e}_2\mathbf{e}_2 \\&= a_1b_1 + a_2b_2 + (a_1b_2 - a_2b_1)\mathbf{e}_1\mathbf{e}_2. \end{align} This can be split into the symmetric, scalar-valued, scalar product and an antisymmetric, bivector-valued exterior product: \begin{align} \mathbf{a} \cdot \mathbf{b} &= a_1b_1 + a_2b_2, \\ \mathbf{a} \wedge \mathbf{b} &= (a_1b_2 - a_2b_1)\mathbf{e}_1\mathbf{e}_2 = (a_1b_2 - a_2b_1)\mathbf{e}_{12}. \end{align} All bivectors in two dimensions are of this form, that is multiples of the bivector , written to emphasise it is a bivector rather than a vector. The magnitude of is , with \mathbf{e}_{12}^2 = -1, so it is called the unit bivector. The term unit bivector can be used in other dimensions but it is only uniquely defined (up to a sign) in two dimensions and all bivectors are multiples of . As the highest grade element of the algebra is also the pseudoscalar which is given the symbol . \theta\mathbf{e}_{12} = i\theta, where is a real number. Putting this into the Taylor series for the exponential map and using the property results in a bivector version of Euler's formula, \exp{\theta\mathbf{e}_{12}} = \exp{i\theta} = \cos{\theta} + i\sin{\theta}, which when multiplied by any vector rotates it through an angle about the origin: (x'\mathbf{e}_1 + y'\mathbf{e}_2) = (x\mathbf{e}_1 + y\mathbf{e}_2) \exp {i\theta}. The product of a vector with a bivector in two dimensions is anticommutative, so the following products all generate the same rotation \mathbf{v}' = \mathbf{v} \exp{i\theta} = \exp(-i\theta)\,\mathbf{v} = \exp({-i\theta}/{2})\,\mathbf{v} \exp({i\theta}/{2}). Of these the last product is the one that generalises into higher dimensions. The quantity needed is called a rotor and is given the symbol , so in two dimensions a rotor that rotates through angle can be written R = \exp({-\tfrac{1}{2} i\theta}) = \exp({-\tfrac{1}{2}\theta\mathbf{e}_{12}}), and the rotation it generates is \mathbf{v}' = R\mathbf{v}R^{-1}. \begin{align} \mathbf{ab} &= (a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3)(b_1\mathbf{e}_1 + b_2\mathbf{e}_2 + b_3\mathbf{e}_3) \\ &= a_1 b_1{\mathbf{e}_1}^2 + a_2 b_2{\mathbf{e}_2}^2 + a_3 b_3{\mathbf{e}_3}^2 + (a_2 b_3 - a_3 b_2)\mathbf{e}_2\mathbf{e}_3 + (a_3 b_1 - a_1 b_3)\mathbf{e}_3\mathbf{e}_1 + (a_1 b_2 - a_2 b_1)\mathbf{e}_1\mathbf{e}_2. \end{align} This can be split into the symmetric, scalar-valued, scalar product and the antisymmetric, bivector-valued, exterior product: \begin{align} \mathbf{a} \cdot \mathbf{b} &= a_1b_1 + a_2b_2 + a_3b_3 \\ \mathbf{a} \wedge \mathbf{b} &= (a_2 b_3 - a_3 b_2)\mathbf{e}_{23} + (a_3 b_1 - a_1 b_3)\mathbf{e}_{31} + (a_1 b_2 - a_2 b_1)\mathbf{e}_{12}. \end{align} In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectors , and form a basis for the space of bivectors , which is itself a three-dimensional linear space. So if a general bivector is: \mathbf{A} = A_{23}\mathbf{e}_{23} + A_{31}\mathbf{e}_{31} + A_{12}\mathbf{e}_{12}, they can be added like vectors \mathbf{A} + \mathbf{B} = (A_{23} + B_{23})\mathbf{e}_{23} + (A_{31} + B_{31})\mathbf{e}_{31} + (A_{12} + B_{12})\mathbf{e}_{12}. while when multiplied they produce the following \mathbf{A} \mathbf{B} = -A_{23}B_{23} - A_{31}B_{31} - A_{12}B_{12} + (A_{12}B_{31} - A_{31}B_{12})\mathbf{e}_{23} + (A_{23}B_{12} - A_{12}B_{23})\mathbf{e}_{31} + (A_{31}B_{23} - A_{23}B_{31})\mathbf{e}_{12} which can be split into symmetric scalar and antisymmetric bivector parts as follows \begin{align} \mathbf{A} \cdot \mathbf{B} &= -A_{12}B_{12} - A_{31}B_{31} - A_{23}B_{23} \\ \mathbf{A} \times \mathbf{B} &= (A_{23}B_{31} - A_{31}B_{23})\mathbf{e}_{12} + (A_{12}B_{23} - A_{23}B_{12})\mathbf{e}_{13} + (A_{31}B_{12} - A_{12}B_{31})\mathbf{e}_{23}. \end{align} The exterior product of two bivectors in three dimensions is zero. A bivector can be written as the product of its magnitude and a unit bivector, so writing for and using the Taylor series for the exponential map it can be shown that \exp{\mathbf{B}} = \exp({\beta\frac{\mathbf{B}}{\beta}}) = \cos{\beta} + \frac{\mathbf{B}}{\beta}\sin{\beta}. This is another version of Euler's formula, but with a general bivector in three dimensions. Unlike in two dimensions bivectors are not commutative so properties that depend on commutativity do not apply in three dimensions. For example, in general in three (or more) dimensions. The full geometric algebra in three dimensions, , has basis ( , , , , , , , ). The element is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors to which they are related, as discussed below. (a + \mathbf{A})(b + \mathbf{B}) = ab + a\mathbf{B} + b\mathbf{A} + \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B}. The even subalgebra, that is the algebra consisting of scalars and bivectors, is isomorphic to the quaternions, . This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates to the negative products in the bivector scalar product . Other quaternion properties can be similarly related to or derived from geometric algebra. This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done the quaternion product is merely the geometric product. It also relates quaternions in three dimensions to complex numbers in two, as each is isomorphic to the even subalgebra for the dimension, a relationship that generalises to higher dimensions. q = \left(\cos{\tfrac {2} \theta}, \omega \sin{\tfrac{1}{2} \theta}\right) In geometric algebra the rotation is represented by a bivector. This can be seen in its relation to quaternions. Let be a unit bivector in the plane of rotation, and let be the angle of rotation. Then the rotation bivector is . The quaternion closely corresponds to the exponential of half of the bivector . That is, the components of the quaternion correspond to the scalar and bivector parts of the following expression: \exp{\tfrac{1}{2} \boldsymbol{\Omega} \theta} = \cos{\tfrac{1}{2} \theta} + \boldsymbol{\Omega}\sin{\tfrac{1}{2} \theta} The exponential can be defined in terms of its power series, and easily evaluated using the fact that squared is . So rotations can be represented by bivectors. Just as quaternions are elements of the geometric algebra, they are related by the exponential map in that algebra. \mathbf{v}' = \exp(-\tfrac{1}{2} \boldsymbol{\Omega} \theta)\,\mathbf{v} \exp(\tfrac{1}{2} \boldsymbol{\Omega} \theta). As in two dimensions, the quantity is called a rotor and written . The quantity is then , and they generate rotations as \mathbf{v}' = R\mathbf{v}R^{-1}. This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions. This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors and represent the same rotation. \mathbf{A} = {\star} \mathbf{a} \,,\quad \mathbf{a} = {\star} \mathbf{A} where is the Hodge star. Note that if the underlying orientation is reversed by inversion through the origin, both the identification of the axial vectors with the usual vectors and the Hodge dual change sign, but the bivectors don't budge. Alternately, using the unit pseudoscalar in , gives \mathbf{A} = \mathbf{a}i \,,\quad \mathbf{a} = - \mathbf{A} i. This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalar squares to , so a negative is needed in one of the products. This relationship extends to operations like the vector-valued cross product and bivector-valued exterior product, as when written as determinants they are calculated in the same way: \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix} \,,\quad \mathbf{a} \wedge \mathbf{b} = \begin{vmatrix} \mathbf{e}_{23} & \mathbf{e}_{31} & \mathbf{e}_{12}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix}\,, so are related by the Hodge dual: {\star} (\mathbf a \wedge \mathbf b ) = \mathbf {a \times b} \,,\quad {\star} (\mathbf {a \times b} ) = \mathbf a \wedge \mathbf b\,. Bivectors have a number of advantages over axial vectors. They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are. For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalise to other dimensions; in particular bivectors can be used to describe quantities like torque and angular momentum in two as well as three dimensions. Also, they closely match geometric intuition in a number of ways, as seen in the next section. |\mathbf{B}| = |\mathbf{a}| |\mathbf{b}|\sin\theta, where is the angle between the vectors. This is the area of the parallelogram with edges and , as shown in the diagram. One interpretation is that the area is swept out by as it moves along . The exterior product is antisymmetric, so reversing the order of and to make move along results in a bivector with the opposite direction that is the negative of the first. The plane of bivector contains both and so they are both parallel to the plane. Bivectors and axial vectors are related as being Hodge dual. In a real vector space, the Hodge dual relates the blade that represents a subspace to its orthogonal complement, so if a bivector represents a plane then the axial vector associated with it is simply the plane's surface normal. The plane has two normal sets of vbectors, one on each side, giving the two possible orientations for the plane and bivector. , torque , linear momentum , and angular momentum .]] In three dimensions, Hodge duality relates the cross product to the exterior product. It can also be used to represent physical quantities, like torque and angular momentum. In vector algebra they are usually represented by pseudovectors that are perpendicular to the plane of the force, linear momentum or displacement that they are calculated from. But if a bivector is used instead, the plane is the plane of the bivector, so is a more natural way to represent the quantities and the way in which they act. Unlike the vector representation, it generalises to other dimensions. The geometric product of two bivectors has a geometric interpretation. For non-zero bivectors and the product can be split into symmetric and antisymmetric parts as follows: \mathbf{AB} = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B}. Like vectors these have magnitudes and , where is the angle between the planes. In three dimensions it is the same as the angle between the normal vectors dual to the planes, and it generalises to some extent in higher dimensions. Bivectors can be added together as areas. Given two non-zero bivectors and in three dimensions it is always possible to find a vector that is contained in both, say, so the bivectors can be written as exterior products involving : \begin{align}\mathbf{B} &= \mathbf{a} \wedge \mathbf{b} \\ \mathbf{C} &= \mathbf{a} \wedge \mathbf{c}\end{align} This can be interpreted geometrically as seen in the diagram: the two areas sum to give a third, with the three areas forming faces of a prism with , , and as edges. This corresponds to the two ways of calculating the area using the distributivity of the exterior product: \begin{align} \mathbf{B} + \mathbf{C} &= \mathbf{a} \wedge \mathbf{b} + \mathbf{a} \wedge \mathbf{c} \\ &= \mathbf{a} \wedge (\mathbf{b} + \mathbf{c}).\end{align} This only works in three dimensions as it is the only number of dimensions in which a vector that is parallel to both bivectors must exist. In a higher number of dimensions, bivectors generally are not associated with a single plane, or if they are (simple bivectors), two bivectors may have no vector in common, and so sum to a non-simple bivector. \mathbf{A} = a_{12}\mathbf{e}_{12} + a_{13}\mathbf{e}_{13} + a_{14}\mathbf{e}_{14} + a_{23}\mathbf{e}_{23} + a_{24}\mathbf{e}_{24} + a_{34}\mathbf{e}_{34}. (\mathbf{e}_{12} + \mathbf{e}_{34})^2 =\mathbf{e}_{12} \mathbf{e}_{12} + \mathbf{e}_{12} \mathbf{e}_{34} + \mathbf{e}_{34} \mathbf{e}_{12} + \mathbf{e}_{34} \mathbf{e}_{34} = -2 + 2 \mathbf{e}_{1234}. The element is the pseudoscalar in , distinct from the scalar, so the square is non-scalar. All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as \mathbf{e}_{12} + \mathbf{e}_{34} = \mathbf{e}_{1} \wedge \mathbf{e}_{2} + \mathbf{e}_{3} \wedge \mathbf{e}_{4}. Similarly, every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover, for a generic bivector the choice of simple bivectors is unique, that is, there is only one way to decompose into orthogonal bivectors; the only exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique. The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero. v' = RvR^{-1}. performing an isoclinic rotation.]] The rotations generated are more complex though. They can be categorised as follows: simple rotations are those that fix a plane in 4D, and rotate by an angle "about" this plane. double rotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique. These are generated by bivectors in a straightforward way. Simple rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors that the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique. Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector , where and are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows: R = \exp(\tfrac{1}{2} (\mathbf{B}_1 + \mathbf{B}_2)) = \exp(\tfrac{1}{2} \mathbf{B}_1)\,\exp(\tfrac{1}{2} \mathbf{B}_2) = \exp(\tfrac{1}{2} \mathbf{B}_2)\,\exp(\tfrac{1}{2} \mathbf{B}_1) It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors. \mathbf{e}_i^2 = \begin{cases} 1, & i = 1, 2, 3 \\ -1, & i = 4 \end{cases} (Note the order and indices above are not universal – here is the time-like dimension). The geometric algebra is , and the subspace of bivectors is . The simple bivectors are of two types. The simple bivectors , and have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, . These bivectors generate ordinary rotations in . The simple bivectors , and have positive squares and as planes span a space dimension and the time dimension. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows: \exp{\tfrac{1}{2}{\boldsymbol{\Omega}\theta}} = \cosh{\tfrac{1}{2} \theta} + \boldsymbol{\Omega}\sinh{\tfrac{1}{2} \theta}, where is the bivector ( , etc.), identified via the metric with an antisymmetric linear transformation of . These are Lorentz boosts, expressed in a particularly compact way, using the same kind of algebra as in and . In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector is of the form R = \exp{\tfrac{1}{2} \mathbf{A}}. The set of all rotations in spacetime form the Lorentz group, and from them most of the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same kind of algebra. \mathbf{F} = \frac{1}{c}\overline{E}\mathbf{e}_4 + \overline{B}\mathbf{e}_{123}, where is again the basis vector for the time-like dimension and is the speed of light. The product yields the bivector that is Hodge dual to in three dimensions, as discussed above, while as a product of orthogonal vectors is also bivector-valued. As a whole it is the electromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the 4-current , a vector quantity given by \mathbf{J} = \overline{j} + c\rho\mathbf{e}_4, where is current density and is charge density. They are related by a differential operator ∂, which is \partial = \nabla - \mathbf{e}_4\frac{1}{c}\frac{\partial}{\partial t}. The operator ∇ is a differential operator in geometric algebra, acting on the space dimensions and given by . When applied to vectors is the divergence and is the curl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity they act as grade lowering and raising differential operators. In particular if is a scalar then this operator is just the gradient, and it can be thought of as a geometric algebraic del operator. Together these can be used to give a particularly compact form for Maxwell's equations with sources: \partial\mathbf{F} = \mathbf{J}. This equation, when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell's four equations. It is also related to the electromagnetic four-potential, a vector given by \mathbf{A} = \overline{A} + \frac{1}{c}V\mathbf{e}_4, where is the vector magnetic potential and is the electric potential. It is related to the electromagnetic bivector as follows \partial\mathbf{A} = -\mathbf{F}, using the same differential operator . \mathbf{p} = \mathbf{A} \lor \mathbf{B} = (\mathbf{A} \times \mathbf{B}) J^{-1}. The operation " " is the meet, which can be defined as above in terms of the join, for non-zero . Using these operations projective geometry can be formulated in terms of geometric algebra. For example, given a third (non-zero) bivector the point lies on the line given by if and only if \mathbf{p} \land \mathbf{C} = 0. So the condition for the lines given by , and to be collinear is (\mathbf{A} \lor \mathbf{B}) \land \mathbf{C} = 0, which in and simplifies to \langle \mathbf{ABC} \rangle = 0, where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1378749384 under CC BY-SA 4.0; modifications have been made.