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# Vectors Position and Displacement by Robert Resnick

One general way of locating a particle (or particle-like object) is with a position vector $\stackrel{\to }{r}$, which is a vector that extends from a reference point (usually the origin) to the particle. In unit-vector notation $\stackrel{\to }{r}$ can be written

$\stackrel{\to }{r}=x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}$ (4-1)

where are the vector components of $\stackrel{\to }{r}$ and the coefficients x, y, and z are its scalar components.

Figure 4-1 The position vector $\stackrel{\to }{r}$ for a particle is the vector sum of its vector components.

The coefficients x, y, and z give the particle's location along the coordinate axes and relative to the origin; that is, the particle has the rectangular coordinates (x, y, z). For instance, Fig. 4-1 shows a particle with position vector

$\stackrel{\to }{r}=\left(-3m\right)\stackrel{^}{i}+\left(2m\right)\stackrel{^}{j}+\left(5m\right)\stackrel{^}{k}$

and rectangular coordinates (-3 m, 2 m, 5 m). Along the x axis the particle is 3 m from the origin, in the $-\stackrel{^}{i}$ direction. Along the y axis it is 2 m from the origin, in the $+\stackrel{^}{J}$ direction. Along the z axis it is 5 m from the origin, in the $+\stackrel{^}{k}$ direction.

As a particle moves, its position vector changes in such a way that the vector always extends to the particle from the reference point (the origin). If the position vector changes - say, from ${\stackrel{\to }{r}}_{1}\text{to}{\stackrel{\to }{r}}_{2}$ during a certain time interval - then the particle's displacement Δ $\stackrel{\to }{r}$ during that time interval is

Δ $\stackrel{\to }{r}={\stackrel{\to }{r}}_{2}-{\stackrel{\to }{r}}_{1}$ (4-2)

Using the unit-vector notation of Eq. 4-1, we can rewrite this displacement as

Δ $\stackrel{\to }{r}=\left({x}_{2}\stackrel{^}{i}+{y}_{2}\stackrel{^}{j}+{z}_{2}\stackrel{^}{k}\right)-\left({x}_{1}\stackrel{^}{i}+{y}_{1}\stackrel{^}{j}+{z}_{1}\stackrel{^}{k}\right)$

or as Δ $\stackrel{\to }{r}=\left({x}_{2}-{x}_{1}\right)\stackrel{^}{i}+\left({y}_{2}-{y}_{1}\right)\stackrel{^}{j}+\left({z}_{2}-{z}_{1}\right)\stackrel{^}{k}$ (4-3)

where coordinates (x1, y1, z1) correspond to position vector $\stackrel{\to }{{r}_{1}}$ and coordinates (x2, y2, z2) correspond to position vector $\stackrel{\to }{{r}_{2}}$. We can also rewrite the displacement by substituting Δx for (x2 x1), Δy for (y2 y1), and Δz for (z2 z1):

Δ$\stackrel{\to }{r}=\mathrm{\Delta x}\stackrel{^}{i}+\mathrm{\Delta y}\stackrel{^}{j}+\mathrm{\Delta z}\stackrel{^}{k}$. (4-4)