Simple Nature - Light and Matter

k/Two vectors, 1, to which

we apply the same operation in

two different frames of reference,

2 and 3.

A useless vector operation example 58

The way I’ve defined the various vector operations above aren’t

as arbitrary as they seem. There are many different vector oper-

ations that we could define, but only some of the possible defini-

tions are mathematically useful. Consider the operation of mul-

tiplying two vectors component by component to produce a third

vector:

Rx=PxQx

Ry=PyQy

Rz=PzQz

As a simple example, we choose vectorsPandQto have length

1, and make them perpendicular to each other, as shown in figure

k/1. If we compute the result of our new vector operation using

the coordinate system shown in k/2, we find:

Rx= 0

Ry= 0

Rz= 0

Thexcomponent is zero becausePx = 0, they component is

zero becauseQy= 0, and thezcomponent is of course zero be-

cause both vectors are in thex-yplane. However, if we carry out

the same operations in coordinate system k/3, rotated 45 degrees

with respect to the previous one, we find

Rx=−1/2

Ry= 1/2

Rz= 0

The operation’s result depends on what coordinate system we

use, and since the two versions ofRhave 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 coordi-

nate system.

All the vector techniques can be applied to any kind of vector,

but the graphical representation of vectors as arrows is particularly

natural for vectors that represent lengths and distances. We define a

vector calledrwhose components are the coordinates of a particular

point in space,x,y, andz. The ∆rvector, whose components are

∆x, ∆y, and ∆z, can then be used to represent motion that starts at

200 Chapter 3 Conservation of Momentum