Figure 3-7 (a) The components ax and ay of vector
$\overrightarrow{a}$.
(b) The components are unchanged if the vector is shifted, as long as the magnitude
and orientation are maintained. (c) The components form the legs of a right triangle whose
hypotenuse is the magnitude of the vector.

Adding vectors geometrically can be tedious. A neater and easier technique involves algebra but requires that the vectors be placed on a rectangular coordinate system. The x and y axes are usually drawn in the plane of the page, in Fig. 3-7a.The z axis comes directly out of the page at the origin; we ignore it for now and deal only with two-dimensional vectors.

A *component* of a vector is the projection of the vector on an axis. In
Fig. 3-7a, for example, ax is the component of vector on (or along) the x axis and
ay is the component along the y axis. To find the projection of a vector along an
axis, we draw perpendicular lines from the two ends of the vector to the axis, as
shown.The projection of a vector on an x axis is its x *component*, and
similarly the projection on the y axis is the y *component*. The process of
finding the components of a vector is called *resolving the vector*.

Figure 3-8 The component of
$\overrightarrow{\mathrm{ba}}$
on the x axis is positive, and that on the y axis is negative.

A component of a vector has the same direction (along an axis) as the vector.
In Fig. 3-7, a_{x} and a_{y} are both positive because
$\overrightarrow{a}$
extends in the positive direction of both axes. (Note the small arrowheads on
the components, to indicate their direction.) If we were to reverse vector
$\overrightarrow{a}$, then both components would be negative and their arrowheads would
point toward negative x and y. Resolving vector
$\overrightarrow{b}$
in Fig. 3-8 yields a positive component b_{x} and a negative component b_{y}.

In general, a vector has three components, although for the case of Fig. 3-7a the component along the z axis is zero. As Figs. 3-7a and b show, if you shift a vector without changing its direction, its components do not change.

**Finding the Components**. We can find the components of
$\overrightarrow{a}$
in Fig. 3-7a geometrically from the right triangle there:

a_{x} = a cos θ and a_{y} + a sin θ, (3-5)

where θ is the angle that the vector makes with the positive direction
of the x axis, and a is the magnitude of
$\overrightarrow{a}$.
Figure 3-7c shows that
$\overrightarrow{a}$
and its x and y components form a right triangle. It also shows how we can
reconstruct a vector from its components: we arrange those components *head*
to tail. Then we complete a right triangle with the vector forming the hypotenuse,
from the tail of one component to the head of the other component.

Once a vector has been resolved into its components along a set of axes, the
components themselves can be used in place of the vector. For example,
$\overrightarrow{a}$
in Fig. 3-7a is given (completely determined) by a and θ. It can also be
given by its components a_{x} and a_{y}. Both pairs of values
contain the same information. If we know a vector in *component notation*
(a_{x} and a_{y}) and want it in *magnitude-angle notation*
(a and θ), we can use the equations

$a=\sqrt{{{a}_{x}}^{}+{{a}_{y}}^{}}$ and tan θ $=\frac{{a}_{y}}{{a}_{x}}$

to transform it.

In the more general three-dimensional case, we need a magnitude and two
angles (say, a, θ, and Θ) or three components (a_{x},
a_{y}, and a_{z}) to specify a vector.

**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."

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