Analytic geometry. Textbook

~c = ~a b such that ~c + b = ~a.

~ ~c = ~a b

~a

~

b

Fig. 10. Di erence of vectors

In the given below de nition of scalar product and in some other cases we use such an agreement.

A product containing probably undetermined factors is set to be zero if at least one of the factors being the part of this product takes the zero value.

~

S ca l a r p rod u c t . For any nonzero vectors ~a and b there is de ned

the angle ' between them satisfying the condition 0 6 ' 6 . To nd this angle we use parallel translation and reduce the vectors to the common initial point. If one of these vectors takes the zero value then this angle is considered undetermined. The scalar product of these vectors is the number

product is also denoted as (~a; b) or ~a b and is called the dot product.

C oo rd i n a t e s o f a v ec t o r . For the points A(a1; a2) and B(b1; b2) on

!

the coordinate plane the coordinates of the vector AB are assumed to be

equal b1 a1 and b2 a2 respectively. This fact is written in the following

!

form: AB = fb1 a1; b2 a2g. The operations with the vectors introduced above may be written in the coordinate form as follows: if is a number,

~a + b = fa1 + b1; a2 + b2g; ~a b = fa1 b1; a2 b2g:

In order to nd conditions of collinearity in the coordinate form, we

introduce the following additional agreement: the equality

means that AD = BC. Being the standard fact in the case when the denominators do not vanish, such an agreement makes it possible to consider the expressions of the form (1) in the cases when B = 0 or D = 0. For

are valid from this point of view. Of course, \fractions" of this kind de ne no numerical values. Such an agreement allows to give the collinearity condition in an easy-to-remember form.

Remark. For such \fractions" the usual transitivity property of the equality relation may be violated. For instance we have the following equal-

ities understood in the sense indicated above

but, of course 12 6= 13. That is why, speaking below (page 20) about the conditions when two equations de ne the same straight line, it is said that the relations (7) are equivalent to three equalities.

plane. These vectors are collinear if and only if at least one of the following

~a 6= 0 and b 6= 0 then these properties take place simultaneously). Hence

~

we get that the vectors ~a and b are collinear if and only if

a1 = a2 : b1 b2

Using the notion of second order determinant, the condition of collinearity

The law of cosines

C

these vectors. We will consider two cases, whether these vectors are collinear or not.

'

~a

Fig. 12.

Simplifying the numerator of the last fraction we get:

=a21 + a22 + b21 + b22 a21 + 2a1b1 b21 a22 + 2a2b2 b22

=2(a1b1 + a2b2):

= a1 a1 +a2 a2 = a1b1 + a2b2:

Corollary. The scalar multiplication has the following properties

c) ( ~a)b = (~ab), ~a( b) = (~ab).

The property a) follows from the de nition of scalar product, the other properties are immediate consequences of the coordinate form of scalar prod-

between them equals =2. The zero vector is considered orthogonal to any vector.

~ ~

We will prove that the ~a and b are orthogonal if and only if ~a b = 0. This

relation is valid if one of the given vectors is the zero vector. In the case

1.2Straight line on the plane

1.2.1Equations of straight lines

The straight line is uniquely determined if we know some its point and a nonzero vector which is orthogonal to this line. Assume that a straight

6 ~

line ` is determined by a point M0 and a vector ~n = 0. The point M is on

!

the line ` if and only if the vectors ~n and M0M are orthogonal (see Fig. 13).

y

N

M ~n

M0

Fig. 13. Equation of the straight line:

the point M is on the line, the point N is not on the line

!

Therefore the equation of the line takes the vector form ~n M0M = 0.

Assume that M0(x0; y0), ~n = fA; Bg. Then for a point M(x; y) we get

!

that M0M = fx x0; y y0g and the equation of the straight line takes the form

or Ax + By (Ax0 + By0) = 0. Denoting C = (Ax0 + By0) we rewrite the obtained equation in the nal form Ax + By + C = 0.

6 ~

Equations of this form are called linear. Since ~n = 0 we have the additional condition jAj + jBj =6 0 (or, equivalently A2 + B2 6= 0). We note that such an equation of the straight line is not unique. The same line is de ned by the equation 2Ax + 2By + 2C = 0, or more generally by the equationsAx + By + C = 0 for arbitrary 6= 01).

The vector ~n de ning the straight line is called its normal vector. It is obvious that it is de ned up to a nonzero scalar factor.

We are going to give conditions when two lines are parallel, orthogonal or coincide.

Assume that we have two straight lines `1 and `2 de ned by the following equations

A1x + B1y + C1 = 0; A2x + B2y + C2 = 0:

Then the following assertion is valid. The lines `1 and `2 are parallel if and

only if A1 = A2, B1 = B2 for some scalar , or equivalently A1 = B1 .

A2 B2

The proof is reduced to the corresponding statement for the normal vectors n1 = fA1; b1g, n2 = fA2; b2g, since the straight lines are parallel if and only if their normal vectors are collinear. Now it remains to apply the given above collinearity condition for the case of vectors.

We apply similar arguments to obtain conditions when the given lines are orthogonal. This property is valid if and only if the normal vectors n1 and n2 are orthogonal.

Using the given above orthogonality condition for vectors we get that the straight lines `1 and `2 are orthogonal if and only if A1A2 + B1B2 = 0.

1) It will be proved below that such equations with arbitrary values of the parameter 6= 0 exhaust all possible linear equations of this straight line.

Now we nd conditions when two lines de ned by the equations written above are equal. We will prove the following assertion.

The lines `1 and `2 coincide if and only if.

for some scalar .

If the conditions (6) are valid then 6= 0, since otherwise A1 = B1 = 0 which is impossible for the equation of the line. In this case the equation of the line `1 may be rewritten in the form

(A2x + B2y + C2) = 0:

Canceling the both sides of this equation by we get the equation of the line `2. Therefore these lines may be de ned by one equation and `1 = `2.

Assume now that `1 = `2. Then these lines are parallel and A1 = A2,

B1 = B2 for some 6= 0. Let M0(x0; y0) be any point on the line. Then

A1 x0 + B1 y0 + C1 = 0; A2x0 + B2y0 + C2 = 0;

20 Chapter 1. Straight lines on the plane

hence C1 = C2, the equalities (6) are valid.

We rewrite the obtained conditions of equality `1 = `2 in the following

We remind that the last condition is the combination of three equalities

Using the second order determinants we reformulate the conditions obtained above.

The straight lines `1 and `2 are parallel if and only if

1.2.2Special cases of equations of straight lines

Here we consider equations of straight lines Ax + By + C = 0 under assumption that some of the coe cients A, B, C vanish.

1) If A = 0 then the equation of the line takes the form By + C = 0. Since B 6= 0 (A and B do not vanish simultaneously), the equation of the line may be rewritten in the form y = BC . Therefore the line is parallel to the axis Ox.

Conversely, if the line is parallel to the x-axis then its equation has the form y = y0 for some constant y0, or the form y y0 = 0. We have an equation of the required type with A = 0.