9781118230725.pdf

3-3 M U LT I P LY I N G V E C T O R S

If the angle between two vectors is 0°, the component of one vector along the

other is maximum, and so also is the dot product of the vectors. If, instead, is 90°,

the component of one vector along the other is zero, and so is the dot product.





The Scalar Product

The scalar productof the vectors and in Fig. 3-18ais written as and
defined to be

 abcosf, (3-20)

whereais the magnitude of ,bis the magnitude of , and is the angle between
and (or, more properly, between the directions of and ). There are actually
two such angles: and 360°. Either can be used in Eq. 3-20, because their
cosines are the same.
Note that there are only scalars on the right side of Eq. 3-20 (including the
value of cos ). Thus on the left side represents a scalarquantity. Because of
the notation, is also known as the dot productand is spoken as “a dot b.”
A dot product can be regarded as the product of two quantities: (1) the mag-
nitude of one of the vectors and (2) the scalar component of the second vector
along the direction of the first vector. For example, in Fig. 3-18b, has a scalar
componentacos along the direction of ; note that a perpendicular dropped
from the head of onto determines that component. Similarly, has a scalar
componentbcos along the direction of . :a

b

:

b

:

:a

b

:



a:

b

:

:a

b

:

 :a

 

b
b a:
:
a:

b 

:

:a

:ab

:

:ab

:

b

:

:a

51

Figure 3-18(a) Two vectors
and , with an angle fbetween
them. (b) Each vector has a
component along the direction
of the other vector.

:b

a:

a

a

b

b

φ

(a)

(b)

Component of b

along direction of

a is b cosφ

Component of a

along direction of

b is a cosφ

φ

Multiplying these gives

the dot product.

Or multiplying these

gives the dot product.

Equation 3-20 can be rewritten as follows to emphasize the components:

 (acosf)(b)(a)(bcosf). (3-21)

The commutative law applies to a scalar product, so we can write

  .

When two vectors are in unit-vector notation, we write their dot product as

 (ax ay az )(bx by bz ), (3-22)

which we can expand according to the distributive law: Each vector component
of the first vector is to be dotted with each vector component of the second vec-
tor. By doing so, we can show that

b axbxaybyazbz. (3-23)

:

:a

b iˆ ˆj kˆ ˆi ˆj kˆ

:

:a

b:a

:

b

:

:a

b

:

:a