Vectors are directed line segments. Two vectors with same origin {bivector} represent a directed plane segment. Bivector attitude, orientation, rotation, or gradient is direction in space of plane-segment normal compared to coordinate axes. Bivector direction, rotation sense, or circulation is sense of rotation of first vector into second vector (clockwise or counterclockwise). Bivector symbol is /| {wedge symbol} between two vectors.
The wedge product of two vectors makes a bivector. Wedge product squared equals negative of first-vector magnitude squared times second-vector magnitude squared times sine of angle A between vectors squared: (a /| b)^2 = - |a|^2 * |b|^2 * sin^2(A). Therefore, bivectors are imaginary numbers, and bivectors are not vectors or scalars.
Bivector magnitude is the parallelogram area: |a| * |b| * sin(A). If bivector magnitude equals zero, vectors are parallel, collinear, or linearly dependent.
Three vectors with same origin {trivector} represent a directed volume segment. Trivector symbol is wedges between vectors: a /| b /| c. k vectors with same origin {k-vector} represent a directed k-volume segment. k vectors can combine to represent a directed k-volume segment. k-vectors can have parallel and perpendicular components.
Cross products of two vectors make a third vector perpendicular to both vectors. Wedge (exterior) products express cross products using only components in the plane made by the two vectors. For two vectors a = x1 * i + x2 * j and b = y1 * i + y2 * j, cross product is (x1*y2 - x2*y1) * k, and wedge product is (x1*y2 - x2*y1) times the ij bivector: (x1*y2 - x2*y1) * i/|j.
Two dimensions have one basis bivector: e12. Three dimensions have three basis bivectors: e23 = i, e31 = j, e12 = k. Three-dimensional bivectors have form a * e23 + b * e31 + c * e12.
Geometric product of a vector and a bivector is interior-product vector plus exterior-product trivector. (Commutator product is zero.) In two-dimensional space, this geometric product rotates the vector. In three-dimensional space, if, for example, vector is a1 * i and bivector is a2 * e23 + b2 * e31 + c2 * e12, geometric product is a1 * i * a2 * e23 + a1 * i * b2 * e31 + a1 * i * c2 * e12 = - a1 * a2 * e123 + + a1 * b2 * j - a1 * c2 * k.
Geometric product of two bivectors is interior-product scalar plus commutator-product bivector plus exterior-product quadrivector. Three-dimensional spaces have no exterior-product quadrivector. For example, if first bivector is a1 * e23 + b1 * e31 + c1 * e12 and second bivector is a2 * e23 + b2 * e31 + c2 * e12, geometric product is - a1 * a2 - b1 * b2 - c1 * c2 + (c1 * b2 - b1 * c2) * e23 + (a1 * c3 - c1 * a3) * e31 + (b1 * a2 - a1 * b2) * e12. Commutator product makes a plane segment. If first bivector is a2 * e23 and second bivector is a2 * e23, geometric product is 0. If first bivector is a1 * e23 and second bivector is b2 * e31, geometric product is a1 * b2 * e23 * e31 = - a1 * b2 * e12.
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Date Modified: 2022.0224