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.
Ordered pairs {Cartesan product} relate first member to second member. Cartesan product is correspondence. Inverse of Cartesan product has bold divide sign.
Vector products {cross product}| {vector product} {outer product} can result in vectors. Cross-product symbol is X: u X v.
magnitude
Cross-product-vector magnitude equals first-vector u length absolute value times second-vector v length absolute value times sine of angle A between vectors: |u| * |v| * sin(A).
direction
Cross-product-vector direction is perpendicular to both original vectors. Cross-product-vector sense is thumb direction if right-hand fingers curl in direction of positive angle between vectors.
coordinates
i-direction coordinate equals first-vector second coordinate x2 times second-vector third coordinate y3 minus first-vector third coordinate x3 times second-vector second coordinate y2. j-direction coordinate equals first-vector third coordinate x3 times second-vector first coordinate y1 minus first-vector first coordinate x1 times second-vector third coordinate y3. k-direction coordinate equals first-vector first coordinate x1 times second-vector second coordinate y2 minus first-vector second coordinate x2 times second-vector first coordinate y1. Therefore, cross-product vector is (x2 * y3 - x3 * y2) * i + (x3 * y1 - x1 * y3) * j + (x1 * y2 - x2 * y1) * k.
Unit-vector cross products make unit vectors. j X k = i. k X i = j. i X j = k. Unit-vector cross products with themselves equal zero: i X i = j X j = k X k = 0.
properties
Cross products are not commutative, because i X j = +k and j X i = -k. i X j = -j X i. i X k = -k X i. j X k = -k X j. Cross products are distributive: c * (i X j) = (c * i) X (c * j) = c * k. Cross products have no inverse, because there is no cross division. Cross products find forces and torques and so curve the function.
Vector products {exterior product} {wedge product} can be cross products of two vectors expressed without using components outside vector plane. If a = x1 * i + x2 * j and b = y1 * i + y2 * j, wedge product is (x1 * y2 - x2 * y1) times bivector of i and j.
For two geometric objects, sum of dot product and wedge product is a product {geometric product} (Clifford). Dot product makes geometric object one grade lower. Wedge product makes geometric object one grade higher.
Wedge products have numbers {grade, vector}| of elements.
Two operations make wedge products {Grassmann product, wedge product}.
For spread right hand, if straight fingers point in same direction as first vector and bent fingers point in same direction as second vector, vector product direction is thumb direction {right hand rule, vector}|. Positive angle is between first and second vector.
Geometric products sum vectors of different dimensions to make a vector type {multivector}.
Vector multiplied by constant {scalar multiplication} makes vector shorter or longer and changes modulus but does not change orientation. Multiplying vector by negative number changes sense, with same orientation but opposite direction. Vector coordinates are distributive over scalar multiplication.
Products {scalar product}| {dot product} {inner product} of two vectors can result in scalars. Scalar product sign is bold dot: u . v, where u and v are vectors.
Scalar equals first vector-length u times second-vector length v times cosine of angle A between vectors: |u| * |v| * cos(A).
Scalar equals first-vector first coordinate x1 times first-vector second coordinate x2 plus second-vector first coordinate y1 times second-vector second coordinate y2: x1 * x2 + y1 * y2.
Both vectors can be the same: x*x + y*y = x^2 + y^2. Two vectors are parallel if they are scalar multiples. Two vectors are perpendicular if their scalar product equals zero.
Scalar product is commutative, is distributive, and has no inverse. Scalar products find energies and so where functions begin or end (boundaries).
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225