Vectors Position and Displacement

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

$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ (4-1)

where $x \hat{i}, y \hat{j}, and z \hat{k}$ are the vector components of $\vec{r}$ and the coefficients x, y, and z are its scalar components.

Figure 4-1 The position vector $\vec{r}$ 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

\vec{r} = (-3 m) \hat{i} + (2 m) \hat{j} + (5 m) \hat{k}

and rectangular coordinates (-3 m, 2 m, 5 m). Along the x axis the particle is 3 m from the origin, in the $- \hat{i}$ direction. Along the y axis it is 2 m from the origin, in the $+ \hat{J}$ direction. Along the z axis it is 5 m from the origin, in the $+ \hat{k}$ 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 ${\vec{r}}_{1} to {\vec{r}}_{2}$ during a certain time interval - then the particle's displacement Δ $\vec{r}$ during that time interval is

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

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

Δ $\vec{r} = (x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}) - (x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k})$

or as Δ $\vec{r} = (x_{2} - x_{1}) \hat{i} + (y_{2} - y_{1}) \hat{j} + (z_{2} - z_{1}) \hat{k}$ (4-3)

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

Δ $\vec{r} = Δx \hat{i} + Δy \hat{j} + Δz \hat{k}$ . (4-4)

About the Authors

David Halliday was an American physicist known for his physics textbooks, Physics and Fundamentals of Physics, which he wrote with Robert Resnick. Both textbooks have been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

The 10th edition of Halliday's Fundamentals of Physics, Extended building upon previous issues by offering several new features and additions. The new edition offers most accurate, extensive and varied set of assessment questions of any course management program in addition to all questions including some form of question assistance including answer specific feedback to facilitate success. The text also offers multimedia presentations (videos and animations) of much of the material that provide an alternative pathway through the material for those who struggle with reading scientific exposition.

Furthermore, the book includes math review content in both a self-study module for more in-depth review and also in just-in-time math videos for a quick refresher on a specific topic. The Halliday content is widely accepted as clear, correct, and complete. The end-of-chapters problems are without peer. The new design, which was introduced in 9e continues with 10e, making this new edition of Halliday the most accessible and reader-friendly book on the market.

A Reader says,"As many reviewers have noted, this is a great physics book used widely in university technical programs as a first course in technical physics, with calculus. I find it is the one book I start with when trying to understand physical concepts at a useful but basic level. It has broad coverage and is well written . To go beyond this book requires specialized books on each topic of interest (electromagnetics, quantum mechanics, thermodynamics, etc.)."

Reader Frank says, "The treatment is sound, thorough, and clear. I've owned the early editions of Halliday and Resnick for years. I'm very happy that I updated my library with this 10th edition. The topics are covered in a very logical order. The study features and worked examples are outstanding. Don't hesitate to buy this book! Reading it is awesome on the Kindle app on the iPad."

Learn more at amazon.com