Analytic geometry. Textbook

assuming that the coordinate of the point Mx on the x-axis equals x and there hold the similar relations for the other axes.

Fig. 1. Cartesian coordinates in the space

Coordinate systems in the space are divided into two classes: righthanded and left-handed.

For right-handed coordinate system the shortest rotation from x-axis to y-axis viewed from the top of z-axis is performed counterclockwise. In the case when the same rotation is clockwise the coordinate system is called left-handed. These cases are illustrated by the following gure.

Fig. 2. Right-handed and left-handed coordinate systems

We always assume that the coordinate system under consideration is right-handed.

3.1.1Distance between two points

Let us consider two points M1(x1; y1; z1) and M2(x2; y2; z2). We take the rectangular parallelepiped whose faces are parallel to the coordinate planes and pass through these two points. The distance between these points equals the diagonal of this parallelepiped.

M1x

M2x

x

Fig. 3. Distance between two points

First of all we remind some well known fact from the school math course.

In the rectangular parallelepiped the square of any diagonal equals the sum of squares of its three edges.

The length of the edge of the parallelepiped which if parallel to the x- axis equals the distance between projections M1x and M2x of the points M1 and M2 on the x-axis. This distance equals jx2 x1j. Similarly we nd the

lengths of the other two edges. They are jy2 y1j and jz2 z1j. From here we get that

p

jM1M2j = (x2 x1)2 + (y2 y1)2 + (z2 z1)2

3.1.2Vectors in the space

Definition. A vector in the space is a directed segment of a straight line, that is, a segment for which it is indicated which of its boundary points

is the beginning and which is the end.

Many de nitions concerning vectors in the space exactly repeat the given earlier de nitions for the case of vectors on the plane. These are the de nitions of equality of vectors, the length of a vector, zero vector, the operations of adding vectors and multiplying a vector by a scalar, the angle between vectors, scalar product and its properties (p. 16), collinear and orthogonal vectors. We do not repeat these de nitions and mention only the properties and relations that are formulated at least slightly di erent di erent from

the case of the plane.

For the points A(a1; a2; a3) and B(b1; b2; b3) in the space the coordinates

scalar product of these vectors is obtained in the coordinate form by the

~

following similar formula: ~ab = a1b1 + a2b2 + a3b3.

pairwise orthogonal. These vectors are called the orthonormal basis in the space.

z

~

k

~|

O

y

~{

x

Fig. 4. Orthonormal basis in the space

Every vector ~a = fa1; a2; a3g has the following representation

~a = a1f1; 0; 0g + a2f0; 1; 0g + a3f0; 0; 1g;

~

so ~a = a1~{ + a2~| + a3k. The obvious geometrical meaning of the values a1, a2, a3 is shown in the following gure.

~

~a

k

y

~{

a1 x

Fig. 5. Coordinates in the space

For an arbitrary vector ~a = fa1; a2; a3g we have the expansion

~

~a = a1~{ + a2~| + a3k:

~

respectively. Then n~{ = cos , ~n~| = cos , ~n k = cos and therefore ~n =

fcos ; cos ; cos g. The values cos , cos , cos are called the direction cosines of the vector ~n. From the equality j~nj = 1 we get that

cos2 + cos2 + cos2 = 1:

An ordered system consisting of three vectors is called the triple. The word \ordered" means that it is known which vector is the rst, which is the

76 Chapter 3. Analytic geometry in the space

second, which is the third. The order of the vectors is usually determined

~

by their position in the record. If the vectors ~a, b, ~c are di erent, then the

following triples obtained by permuting the vectors will also be di erent:

is a plane to which they are all parallel.

Further we'll use the following obvious properties:

•Any two vectors are coplanar.

•The zero vector is parallel to all planes.

In the case of coplanar vectors we'll assume that these vectors are reduced to one initial point and the corresponding plane (not necessarily unique) passes through this point.

Definition. The triple of non-coplanar vectors is called right-handed if after reducing these vectors to the common initial point, the shortest

~

rotation from ~a to b viewed from the ending point of ~c onto the plane

~

de ned by ~a and b is performed counterclockwise. If the same rotation is performed clockwise the triple is called left-handed.

Remark. It is obvious that a coordinate system is right-handed if and

~

only if the corresponding vectors ~{, ~|, k form the right-handed triple and

that the similar fact is valid for the case of left-handed coordinate systems.

1) j~c j = j~a j jb j sin ' where ' is the angle between ~a and b;

j j j~ j

~a b sin ' at least one of the rst two factors vanishes, therefore the

product takes the zero value independently of the undetermined term

• ~a 6= 0 and b 6= 0, then ' = 0 or ' = , sin ' = 0, ~c = 0 as in the previous case.

~

Therefore, in the case of collinear vectors ~a and b, vector ~c is uniquely

de ned by property 1), property 2) holds, and property 3) does not need to be checked.

jbj 6= 0, the angle ' is de ned, 0 < ' < , therefore j~c j 6= 0. Adding

the property 2) we see that there are de ned exactly two vectors since this

~

vector is orthogonal to ~a and b and has the obtained before nonzero length.

Now the property 3) permits to select one and only one vector from these two ones.

So in each case the vector product is uniquely determined. The vector

also called cross product and is sometimes denoted as [~a b] or [~a; b].

~

In the following, we use some properties of triples. A triple ~a, b, ~c of non-

collinear vectors is either right-handed, or left-handed. These properties we will call the type of a triple. Speaking about transformations of triples we say that the transformation under consideration does not change (changes) the type of a triple if from a triple of some type we always obtain the triple of the same (of the opposite) type.

This property immediately follows the de nition of rightand left-handed triples. Moreover, swapping any two vectors of a triple changes its type. It is easy to see that the cyclic rearrangements of vectors in the triple (obtaining

If > 0 then the triples ~a, b, ~c and ~a, b, ~c have the same type since

~

the shortest rotations from ~a to b when viewed from the ending point of ~c

or when viewed from the similar point of ~c are performed in the same direction. This property is true for any vector of a triple: multiplication of any vector of a triple by a positive scalar does not change the type of this triplet.

If we change the sign of one vector of a triple then the triple changes its type. Combining this property with the previous one, we get that multiplication of any vector of a triplet by a negative scalar changes the type of

this triplet.

Now we turn to the properties of vector product. From the analysis following the de nition of vector product we obtain the following property.

~

orthogonal to each of the vectors ~a and b. After applying the property 2)

~ from the de nition we get that there are only two possible values for b ~a:

~a, > 0

~a

~

' b

~a, < 0

Similar arguments permit to obtain the equality (~a b) = ~a ( b).

Remark. It is assumed that the operation of multiplication by a scalar has higher priority than the operation of vector multiplication, and the

~

vector ~c = ~a b equals the area of the parallelogram with adjacent sides ~a

vectors.

~

b

h

'

~a

Fig. 7. Parallelogram de ned by two vectors

The base of the parallelogram equals a, the height of the parallelogram

j ~j h = b sin ', the area A = ah = ab sin ' = ~a b .

Now we are going to nd the vector product in the coordinate form.

~

For the vectors ~{, ~|, k from the orthonormal basis using the de nition of

1)In the case < 0 multiplying the rst vector by we change the type of the triple, then multiplication

of the third vector by restores the initial type.