Editing Euclidean vector (section)

=== Opposite, parallel, and antiparallel vectors {{anchor|antiparallel|opposite|parallel}}===

Two vectors are ''opposite'' if they have the same magnitude but [[opposite direction (geometry)|opposite direction]];<ref name=HMCS/> so two vectors

<math display=block>{\mathbf a} = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3</math>

and

<math display=block>{\mathbf b} = b_1{\mathbf e}_1 + b_2{\mathbf e}_2 + b_3{\mathbf e}_3</math>

are opposite if

<math display=block>a_1 = -b_1,\quad a_2=-b_2,\quad a_3=-b_3.\,</math>

Two vectors are ''[[equidirectional]]'' (or ''codirectional'') if they have the same direction but not necessarily the same magnitude.<ref name=HMCS/>
Two vectors are ''parallel'' if they have either the same or opposite direction, but not necessarily the same magnitude; two vectors are ''antiparallel'' if they have strictly opposite direction, but not necessarily the same magnitude.{{efn|"Can be brought to the same straight line by means of parallel displacement".<ref name=HMCS>{{cite book |last1=Harris |first1=John W. |last2=Stöcker |first2=Horst |year=1998 |title=Handbook of mathematics and computational science |publisher=Birkhäuser |isbn=0-387-94746-9 |at=Chapter 6, p. 332 |url=https://books.google.com/books?id=DnKLkOb_YfIC&pg=PA332 }}</ref>}}