Vector Subtraction and Position Vectors

vector subtraction is defined by addition of the negative:

$\displaystyle \mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v}).$

Equivalently, $\mathbf{u}-\mathbf{v}$ is the vector which, when added to $\mathbf{v}$ , gives $\mathbf{u}$ .

Figure 5a, modernized: vector addition and subtraction.

If $O$ is a fixed origin, define the position vectors

$\displaystyle \mathbf{r}_A=\overrightarrow{OA},\qquad \mathbf{r}_B=\overrightarrow{OB}.$

Then the displacement from $A$

$\displaystyle \boxed{\overrightarrow{AB}=\mathbf{r}_B-\mathbf{r}_A.} \tag{1}$

This relation is fundamental in mechanics: displacement is the difference of position vectors.

Figure 5b, modernized: displacement as the difference of position vectors.

Modern notation references

The notation and terminology in this modernized article follow standard present-day mechanics and vector-analysis usage, particularly:

J. R. Taylor, classical mechanics, University Science Books, 2005.
D. Kleppner and R. Kolenkow, An Introduction to Mechanics, 2nd ed., Cambridge University Press, 2014.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison–Wesley, 2002.

Source

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

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