the dot product of two vectors is
(the vectors arent column vectors here, they are written as such to make the operation easier to grasp.)
the dot product of two vectors can be written as the matrix product of the transpose of one vector by the other:
let be vectors in or in , and let be the angle between them. then
the scalar product of 2 vectors A and B is defined as where is the angle between A and B when they are drawn tail to tail (to eliminate ambiguity, is always taken as the angle smaller than ):
when the vectors are in the form of a list of components, e.g. then the dot product is the sum of the products of corresponding components, i.e. given two vectors , their dot product is
some stuff from college
find a unit vector in the plane which is perpendicular to
we denote the perpendicular vector by , since is in the plane, , for B to be perpendicular to A, we have because , so:
hence , however, B is a unit vector, which means that combining these gives , or and the ambiguity in sign of and indicates that B can point along a line perpendicular to A in either of two directions