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.
Mathematical Sciences>Calculus>Vector>Operations>Product
3-Calculus-Vector-Operations-Product
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Date Modified: 2022.0224