Calculus: Analytic Geometry and Calculus, with Vectors

34 Analytic geometry in two dimensions

Finally, tell why (5) must hold. Remark: In order to appreciate the significance
of this work, we must do a little thinking about "elementary" mathematics.
It is sometimes said that a straight line is the shortest distance between two points.
If this silly collection of words means anything it means that the length (a num-
ber) of the line segment (a point set) joining two points P1 and P2 is less than the
length (a number) of each other path (a point set) joining P1 and P2. We must
study more mathematics before we can learn what we mean by a path joining
Pi and P2 and what we mean by the length of such a path. In some parts of
"advanced" mathematics, the multifarious axioms of Euclid and the theorem
of Pythagoras are bypassed and the number d in the formula

d = (x2 - x1)2 + (y2 - y1)2

is defined to be the distance (in Euclid space of two dimensions) between the

two points P1(x1iy1) and P2(x2,y2). It is useful as well as possible to define d by

other formulas to obtain spaces that are not Euclid spaces. In such situations

it is necessary to use analytical methods instead of geometrical methods to

determine whether triangle inequalities hold.

33 Four numbers all, all, all, and all determine the equations

(1) x'=a11x+a12y

Y' = a21x + a22y

into which we can substitute the coordinates of a given point (x,y) to obtain the

coordinates of a transform, or transformed point, (x',y'). Supposing that (x1,y1)

and (x2,y2) are two given points and that D is the distance between their trans-

forms (xl, yl) and (x2i y2), find D2. Ans.:

(2) D2 = (ail + a21)(x2 - x1)2 + (ail + a22)(Y2 - y1)2

+ 2(ana12 + a21a22)(x2 - x1)(Y2 - y1)

Remark: The transformer is called isometric if the distance d between two points

is always the same as the distance D between their transforms. If the trans-

former is isometric, we can put x2- xl = 1 and y2 - y1 = 0 to obtain

(3) ail + a21 = 1,

we can put x2 - x1 = 0 and y2 - y1 = 1 to obtain

(4) ail + a22 = 1,

and we can put x2 - x1 = 1 and y2- y1 = 1 and use (3) and (4) to obtain

(5) a11a12 + a21a22 = 0.

On the other hand, if (3), (4), and (5) hold, then (2) shows that the transformer

is isometric.

(^34) Supposing that the first of the two equations
(1) Ix+By= - C, Bx - Xy = Bxo - Iyo
is the equation of a given line L and that Po(xo,yo) isa given point, find the