9781118230725.pdf

50 CHAPTER 3 VECTORS

Multiplying Vectors*

There are three ways in which vectors can be multiplied, but none is exactly like

the usual algebraic multiplication. As you read this material, keep in mind that a

vector-capable calculator will help you multiply vectors only if you understand

the basic rules of that multiplication.

Multiplying a Vector by a Scalar

If we multiply a vector by a scalar s, we get a new vector. Its magnitude is

the product of the magnitude of and the absolute value of s. Its direction is the

direction of if sis positive but the opposite direction if sis negative. To divide

bys, we multiply by 1/s.

Multiplying a Vector by a Vector

There are two ways to multiply a vector by a vector: one way produces a scalar

(called the scalar product), and the other produces a new vector (called the vector

product). (Students commonly confuse the two ways.)

:a

:a :a

a:

:a

Key Ideas

●The vector (or cross) product of two vectors and is

written  and is a vector whose magnitude cis given by

cabsin,

in which is 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 (  ). In unit-vector

notation,

 

which we may expand with the distributive law.

●In nested products, where one product is buried inside an-

other, follow the normal algebraic procedure by starting with

the innermost product and working outward.

b (axˆiayjˆazkˆ ) (bxˆibyjˆbzkˆ ),

:

:a

b :a

:

b

:

:a

b

:

a:

b :c

:

:a

b :c

:

:a

b

:

:a

*This material will not be employed until later (Chapter 7 for scalar products and Chapter 11 for vec-

tor products), and so your instructor may wish to postpone it.

●The product of a scalar sand a vector is a new vector
whose magnitude is and whose direction is the same as
that of if sis positive, and opposite that of if sis negative.
To divide by s, multiply by 1/s.

●The scalar (or dot) product of two vectors and is writ-
ten  and is the scalarquantity given by

 abcos,

in which is the angle between the directions of and.
A scalar product is the product of the magnitude of one vec-
tor and the scalar component of the second vector along the
direction of the first vector. In unit-vector notation,

 

which may be expanded according to the distributive law.
Note that  b:a.
:
b
:
a:

b (axˆiayjˆazkˆ )(bxˆibyjˆbzˆk ),

:

:a

b

:

:a

b

:

:a

b

:

:a

b

:

a:

v: v:

:v v:

sv

v:

3-3MULTIPLYING VECTORS

Learning Objectives

3.13Given two vectors, use a dot product to find how much

of one vector lies along the other vector.

3.14Find the cross product of two vectors in magnitude-

angle and unit-vector notations.

3.15Use the right-hand rule to find the direction of the vector

that results from a cross product.

3.16In nested products, where one product is buried inside

another, follow the normal algebraic procedure by starting

with the innermost product and working outward.

After reading this module, you should be able to...

3.09Multiply vectors by scalars.

3.10Identify that multiplying a vector by a scalar gives a vec-

tor, taking the dot (or scalar) product of two vectors gives a

scalar, and taking the cross (or vector) product gives a new

vector that is perpendicular to the original two.

3.11Find the dot product of two vectors in magnitude-angle

notation and in unit-vector notation.

3.12Find the angle between two vectors by taking their dot prod-

uct in both magnitude-angle notation and unit-vector notation.