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3
CHAPTER
Vectors and
Two-Dimensional Motion
OUTLINE
3.1
Vectors and Their
Properties
3.2
Components of a Vector
3.3
Displacement, Velocity,
and Acceleration in Two
Dimensions
3.4
Motion in Two Dimensions
3.5
Relative Velocity
In our discussion of one-dimensional motion in Chapter 2, we used the concept of vectors
only to a limited extent. In our further study of motion, manipulating vector quantities will be-
come increasingly important, so much of this chapter is devoted to vector techniques. We’ll
then apply these mathematical tools to two-dimensional motion, especially that of projectiles,
and to the understanding of relative motion.
3.1
VECTORS AND THEIR PROPERTIES
Each of the physical quantities we will encounter in this book can be categorized as
either a
vector quantity
or a
scalar quantity
. As noted in Chapter 2, a vector has both
direction and magnitude (size). A scalar can be completely speciFed by its magni-
tude with appropriate units; it has no direction. An example of each kind of quan-
tity is shown in ±igure 3.1 (page 54).
As described in Chapter 2, displacement, velocity, and acceleration are vector
quantities. Temperature is an example of a scalar quantity. If the temperature of
an object is
±
5
8
C, that information completely speciFes the temperature of the ob-
ject; no direction is required. Masses, time intervals, and volumes are scalars as
well. Scalar quantities can be manipulated with the rules of ordinary arithmetic.
Vectors can also be added and subtracted from each other, and multiplied, but
there are a number of important differences, as will be seen in the following
sections.
When a vector quantity is handwritten, it is often represented with an arrow
over the letter ( ). As mentioned in Section 2.1, a vector quantity in this book
A
:
Legendary motorcycle stuntman Evel
Knievel blasts off in his custom
rocket-powered Harley-Davidson
Skycycle in an attempt to jump the
Snake River Canyon in 1974. A
parachute prematurely deployed and
caused the craft to fall into the
canyon, just short of the other side.
Knievel survived.
© Bettmann/Corbis

54
Chapter 3
Vectors and Two-Dimensional Motion
will be represented by boldface type with an arrow on top (for example,
). The
magnitude of the vector
will be represented by italic type, as
A
. Italic type will
also be used to represent scalars.
Equality of Two Vectors.
Two vectors
and
are equal if they have the same
magnitude and the same direction. This property allows us to translate a vector
parallel to itself in a diagram without affecting the vector. In fact, for most
purposes, any vector can be moved parallel to itself without being affected. (See
Fig. 3.2.)
Adding Vectors.
When two or more vectors are added, they must all have the
same units. For example, it doesn’t make sense to add a velocity vector, carrying
units of meters per second, to a displacement vector, carrying units of meters.

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