Two free vectors are equal when they have the same magnitude and the same direction. Thus

$\displaystyle \mathbf{a}=\mathbf{b}$

means that one can translate either directed segment parallel to itself until it coincides with the other.

For a parallelogram $ABCD$ ,

$\displaystyle \overrightarrow{AB}=\overrightarrow{DC}, \qquad \overrightarrow{AD}=\overrightarrow{BC}.$

This is the geometric content behind treating a free vector independently of where it is drawn.

For localized vectors, equality as free vectors does not by itself imply the same mechanical effect. More advanced mechanics distinguishes the vector value from its point of application or line of action.

Source

This article is a modernized restatement of the Public domain corresponding article in Louis Brand, Vectorial Mechanics, John Wiley & Sons, New York, 1930, Chapter I, “Vector Algebra.” The original 1930 edition is the source basis.

"equality of vectors" is owned by bloftin.
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