138 direction cosines

)

f

()

)

Although Diophantus did not use sophisticated alge-

braic notation, he was the first to use a symbol for an

unknown quantity and to introduce a notation for

powers of that unknown. He also used an abbreviation

for the word equals. This represents the first step in

history toward moving from verbal algebra to sym-

bolic algebra.

Diophantus’s text was profoundly influential and,

centuries later, was deemed essential reading for Euro-

pean scholars of the Renaissance. Inspired by an exer-

cise in the text, scholar P

IERRE DE

F

ERMAT

(1601–65)

scrawled the famous comment in the margin of his per-

sonal copy of Arithmetica that spurred three centuries

of intense mathematical research in number theory. This

comment became known as F

ERMAT

’

S LAST THEOREM

.

direction cosines Each point Pon the surface of a

unit sphere determines a unique direction in three-

dimensional space: if Ois the center of the sphere, then

the ray connecting Oto Pspecifies a direction. Con-

versely, the direction of any given line in space corre-

sponds to a point Pon the unit sphere.

Setting Oto be the origin of a C

ARTESIAN COORDI

-

NATE

system, the “direction cosines” of any directed

line in three-dimensional space are simply the coordi-

nates of the point Pon the unit sphere that corresponds

to the direction of that line. For example, the direction

cosine of the positive x-axis is (1,0,0), and that of the

negative z-axis is (0,0,–1).

The use of the word cosine in the name of this con-

cept comes from the observation that the direction of a

line through Ois completely specified by the three

angles α, β, and γit makes with each of positive the x-,

y-, and z-axes, respectively. (These angles are assumed

to lie between zero and 180°. They are called the direc-

tion angles.) An exercise in geometry then shows that

the corresponding point Pon the unit sphere has coor-

dinates (cos α, cos β, cos γ).

The three direction cosines are not independent.

Two applications of P

YTHAGORAS

’

S THEOREM

show

that these numbers satisfy the relation: cos2α+ cos2β

+ cos2γ= 1. Thus any two direction cosines determine

the third.

The direction cosines of an arbitrary line are often

denoted (l,m,n).The “direction ratios” or “direction

numbers” of a line are defined as any set of three num-

bers in the ratio l : m : n. The angle θbetween two

lines with direction cosines (l1,m1,n1) and (l2,m2,n2) is

given by:

cos θ= l1l2+ m1m2+ n1n2

This is simply the

DOT PRODUCT

of the two

VECTOR

s

that describe the directions of the lines.

directional derivative The graph of a function z=

f(x,y) is a surface sitting in three-dimensional space.

The directional derivative of fat a point P= (x,y) and

in the direction given by a

VECTOR

v= < v1, v2>,

denoted Dvf, is simply the

SLOPE

of the surface above

the point Pin the direction of v. It is assumed that vis

a vector of length 1.

Specifically, if tis a variable, best thought of as

“time,” then the expression P+ tvrepresents a straight-

line path starting at Ppointing in the direction of v,

and f(P+ tv) is the “slice” of the surface above this

line. The directional derivative is then the

DERIVATIVE

of this quantity with respect to t:

(We require vto be a vector of unit length so that the

“speed” at which we traverse the path P+ tvis 1 unit

of length per unit time.)

If we take vto be the unit vector in the direction

of the positive x-axis, v= (1,0), then

, the

PARTIAL DERIVATIVE

of the

function with respect to x. Similarly, the directional

derivative in the direction of the positive y-axis is the

partial derivative with respect to y. In general, the

CHAIN

RULE

shows: