difference of vectorsDifferenceOfVectors2008-02-09 13:14:032008-08-26 06:28:17Definitionparallelogram principledifference vectoropposite vector% this is the default PlanetMath preamble. as your knowledge
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Let $\vec{a}$ and $\vec{b}$ be two vectors in the plane (or in a vector space).\, The {\em difference vector} or {\em difference} \,$\vec{a}\!-\!\vec{b}$\, of $\vec{a}$ and $\vec{b}$ is a vector $\vec{d}$\, such that
$$\vec{b}+\vec{d} = \vec{a}.$$
Thus we have
\begin{align}
\vec{b}+(\vec{a}\!-\!\vec{b}) = \vec{a}.
\end{align}
According to the procedure of forming the sum of vectors by setting the addends one after the other, the equation (1) tallies with the picture below; when the minuend and the subtrahend emanate from a common initial point, their difference vector can be directed from the terminal point of the subtrahend to the terminal point of the minuend.
\begin{center}
\begin{pspicture}(0,-1)(4,2.5)
\psdot[linecolor=black](0,0)
\psline[arrows=->,arrowsize=5pt,linecolor=black](0,0)(-1.5,2)
\psline[arrows=->,arrowsize=5pt,linecolor=red](-1.5,2)(4,1)
\psline[arrows=->,arrowsize=5pt,linecolor=blue](0,0)(4,1)
\rput[a](2.1,0.2){$\vec{a}$}
\rput[a](-1,0.7){$\vec{b}$}
\rput[a](1.6,1.76){$\vec{a}\!-\!\vec{b}$}
\end{pspicture}
\end{center}
\textbf{Remark.}\, It is easily seen that the difference $\vec{a}\!-\!\vec{b}$ is same as the sum vector
$$\vec{a}\!+\!(-\vec{b})$$
where $-\vec{b}$ is the {\em opposite vector} of $\vec{b}$:\, it may be represented by the directed line segment from the terminal point of $\vec{b}$ to the initial point of $\vec{b}$.