stylistic:tone
Equality

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. 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 equal 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}.\,}