stylistic:tone, structural:expansion, information:extraction, information:highlighting
Opposite, parallel, and antiparallel vectors

Two vectors are opposite if they have the same magnitude but opposite direction; so two vectors

a = a 1 e 1 + a 2 e 2 + a 3 e 3 {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}

and

b = b 1 e 1 + b 2 e 2 + b 3 e 3 {\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}

are opposite if

a 1 = − b 1, a 2 = − b 2, a 3 = − b 3. {\displaystyle a_{1}=-b_{1},\quad a_{2}=-b_{2},\quad a_{3}=-b_{3}.\,}

Two vectors are equidirectional (or codirectional) if they have the same direction but not necessarily the same magnitude. Two vectors are parallel if they have the same or opposite direction but not necessarily the same magnitude.