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Vector Subtraction

Figure 5.4 illustrates the vector difference $ \underline{w}=\underline{x}-\underline{y}$ between $ \underline{x}=(2,3)$ and $ \underline{y}=(4,1)$. From the coordinates, we compute $ \underline{w}= \underline{x}-\underline{y}= (-2, 2)$.

Figure 5.4: Geometric interpretation of a difference vector.
\includegraphics[scale=0.7]{eps/vecsub}
Note that the difference vector $ \underline{w}$ may be drawn from the tip of $ \underline{y}$ to the tip of $ \underline{x}$ rather than from the origin to the point $ (-2,2)$; this is a customary practice which emphasizes relationships among vectors, but the translation in the plot has no effect on the mathematical definition or properties of the vector. Subtraction, however, is not commutative. To ascertain the proper orientation of the difference vector $ \underline{w}=\underline{x}-\underline{y}$, rewrite its definition as $ \underline{x}=\underline{y}+\underline{w}$, and then it is clear that the vector $ \underline{x}$ should be the sum of vectors $ \underline{y}$ and $ \underline{w}$, hence the arrowhead is on the correct endpoint. Or remember ``$ x-y$ points to $ x$,'' or ``$ x-y$ is $ x$ from $ y$.''
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