dot product
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.)
[cite:;taken from @calc_hubbard_2015 definition 1.4.1]
of two vectors is(the vectors arent column vectors here, they are written as such to make the operation easier to grasp.)[cite:;taken from @calc_hubbard_2015 definition 1.4.1]
the dot product of two vectors can be written as the matrix product of the transpose of one vector by the other:
[cite:@calc_hubbard_2015 chapter 1 vectors, matrices, and derivatives]
[cite:@calc_hubbard_2015 chapter 1 vectors, matrices, and derivatives]let 
be vectors in 
or in 
, and let 
be the angle between them. then
[cite:;taken from @calc_hubbard_2015 proposition 1.4.3 geometric interpretation of the dot product]
be vectors in or in , and let be the angle between them. then[cite:;taken from @calc_hubbard_2015 proposition 1.4.3 geometric interpretation of the dot product]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 
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 
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
, 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