156

7.4*Unit Vector Notation

When we want to specify a vector by its components, it can be cumber-

some to have to write the algebra symbol for each component:

.

x = 290 km,

.

y = 230 km

A more compact notation is to write

.

r = (290 km)

x

+ (230 km)

y

,

where the vectors

x

,

y

, and

z

, called the unit vectors, are defined as the

vectors that have magnitude equal to 1 and directions lying along the x, y,

and z axes. In speech, they are referred to as “x-hat” and so on.

A slightly different, and harder to remember, version of this notation is

unfortunately more prevalent. In this version, the unit vectors are called

i

,

j

, and

k

:

.

r = (290 km)

i

+ (230 km)

j

.

7.5*Rotational Invariance

Let’s take a closer look at why certain vector operations are useful and

others are not. Consider the operation of multiplying two vectors compo-

nent by component to produce a third vector:

R

x

=P

x

Q

x

R

y

=P

y

Q

y

R

z

=P

z

Q

z

As a simple example, we choose vectors P and Q to have length 1, and

make them perpendicular to each other, as shown in figure (a). If we

compute the result of our new vector operation using the coordinate system

shown in (b), we find:

R

x

=0

R

y

=0

R

z

=0

The x component is zero because P

x

=0, the y component is zero because

Q

y

=0, and the z component is of course zero because both vectors are in the

x-y plane. However, if we carry out the same operations in coordinate

system (c), rotated 45 degrees with respect to the previous one, we find

R

x

=1/2

R

y

=–1/2

R

z

=0

The operation’s result depends on what coordinate system we use, and since

the two versions of R have different lengths (one being zero and the other

nonzero), they don’t just represent the same answer expressed in two

different coordinate systems. Such an operation will never be useful in

physics, because experiments show physics works the same regardless of

which way we orient the laboratory building! The useful vector operations,

such as addition and scalar multiplication, are rotationally invariant, i.e.

come out the same regardless of the orientation of the coordinate system.

Chapter 7Vectors

P

Q

x

y

x

y

(a)

(b)

(c)