9781118230725.pdf

REVIEW & SUMMARY 55

Scalars and Vectors Scalars,such as temperature, have magni-

tude only. They are specified by a number with a unit (10°C) and

obey the rules of arithmetic and ordinary algebra.Vectors,such as

displacement, have both magnitude and direction (5 m, north) and

obey the rules of vector algebra.

Adding Vectors Geometrically Two vectors and may

be added geometrically by drawing them to a common scale

and placing them head to tail. The vector connecting the tail of

the first to the head of the second is the vector sum. To

subtract from , reverse the direction of to get  ; then

add to. Vector addition is commutative

and obeys the associative law

.

Components of a Vector The (scalar) components axandayof

any two-dimensional vector along the coordinate axes are found

by dropping perpendicular lines from the ends of onto the coor-

dinate axes. The components are given by

axacosu and ayasinu, (3-5)

whereuis the angle between the positive direction of the xaxis

and the direction of. The algebraic sign of a component indi-

cates its direction along the associated axis. Given its compo-

nents, we can find the magnitude and orientation (direction) of

the vector by using

and

Unit-Vector Notation Unit vectors, , and have magnitudes of

unity and are directed in the positive directions of the x, y,andz

axes, respectively, in a right-handed coordinate system (as defined

by the vector products of the unit vectors). We can write a vector

in terms of unit vectors as

axay az , (3-7)

in which ax,ay, and az are the vector componentsof and ax,ay,

andazare its scalar components.

iˆ jˆ kˆ a:

:a iˆ jˆ kˆ

:a

iˆjˆ kˆ

a 2 a^2 xa^2 y

a:

a:

:a

:a

(a:b

:

):c:a(b

:

:c)

:ab

:

b

:

:a

b :a

: b

:

b

:

b a:

: s

:

b

:

a:

Review & Summary

Adding Vectors in Component Form To add vectors in com-

ponent form, we use the rules

rxaxbx ryayby rzazbz. (3-10 to 3-12)

Here and are the vectors to be added, and is the vector sum.

Note that we add components axis by axis.We can then express the

sum in unit-vector notation or magnitude-angle notation.

Product of a Scalar and a Vector The product of a scalar sand

a vector is a new vector whose magnitude is svand whose direc-

tion is the same as that of if sis positive, and opposite that of if

sis negative. (The negative sign reverses the vector.) To divide by

s, multiply by 1/s.

The Scalar Product The scalar(ordot)productof two vectors

and is written  and is the scalarquantity given by

 abcosf, (3-20)

in which fis the angle between the directions of and. A scalar

product is the product of the magnitude of one vector and the

scalar component of the second vector along the direction of the

first vector. Note that    which means that the scalar

product obeys the commutative law.

In unit-vector notation,

 (axay az )(bxby bz ), (3-22)

which may be expanded according to the distributive law.

The Vector Product The vector(orcross)productof two vectors

and is written  and is a vector whose magnitude cis

given by

cabsinf, (3-24)

in which fis the smaller of the angles between the directions of

and. The direction of is perpendicular to the plane

defined by and and is given by a right-hand rule, as shown in

Fig. 3-19. Note that (  ), which means that the vec-

tor product does not obey the commutative law.

In unit-vector notation,

(axay az )(bxby bz ), (3-26)

which we may expand with the distributive law.

b iˆ ˆj kˆ ˆi ˆj kˆ

:

:a

b :a

:

b

:

:a

b

:

a:

b c:

: a

:

b :c

:

b a:

:

:a

b iˆ ˆj kˆ ˆi ˆj kˆ

:

a:

b:a,

:

b

:

:a

:a b:

:ab:

:b :ab:

a:

v:

:v

v: :v

v:

b :r

:

:a

Additional examples, video, and practice available at WileyPLUS

We next evaluate each term with Eq. 3-24, finding the

direction with the right-hand rule. For the first term here,

the angle fbetween the two vectors being crossed is 0. For

the other terms,fis 90°. We find

6(0)9()8( ) 12

 12 iˆ 9 jˆ8 .kˆ (Answer)

:c jˆ kˆ iˆ

This vector is perpendicular to both and , a fact youb

:

:c :a

can check by showing that  = 0 and  = 0; that is, there

is no component of along the direction of either or.

In general: A cross product gives a perpendicular

vector, two perpendicular vectors have a zero dot prod-

uct, and two vectors along the same axis have a zero

cross product.

b

:

:c :a

b

:

:c:a :c

(3-2)

(3-3)

tan

ay

ax

(3-6)