16. Basis in the geometric space (plane and line). The coordinates of a vector in a basis.
A basis on a line is any non-zero vector belonging to this line. A basis on a plane is any pair of linearly independent vectors belonging to the plane. A basis on space is any triple of linearly independent vectors.
A basis is orthogonal if the vectors forming the basis are pairwise orthogonal (mutually perpendicular). An orthogonal basis is orthonormal if the vectors forming the basis have the length 1.
A
spatial basis composed of linearly independent vectors
is denoted by
.
An orthogonal or orthonormal basis is denoted by
.
Theorem.
Let
be a basis. Then every vector
in the space can be uniquely represented as
where
are some numbers.
Theorem.
Two vectors
and
on
plane are linearly dependent iff their coordinates in some basis
satisfy the condition
.
Proof.
()
Let
and
be linearly dependent. Then
or in coordinate form
.
Excluding
from these two scalar equalities, we have
,
i.e.
.
()
Let
.
Then we have
for
and
,
i.e. the corresponding coordinates of vectors
and
are proportional and consequently
and
are linearly dependent. The case
is proposed to consider yourself.
Theorem.
Three vectors
,
and
in space are linearly dependent iff their coordinates in some basis
satisfy the condition
.
Corollary. The equalities and are necessary and sufficient conditions of collinearity of a pair of vectors on plane and coplanarity of a triple of vectors in space respectively.
17. Cartesian system of coordinates. Radius-vector of a point. Finding the coordinates of a point dividing a segment in some ratio.
The
set consisting of a basis
and a point O
in which are put the beginnings of all basis vectors is called a
common
Cartesian system of coordinates
and is denoted by
.
A system of coordinates
generated by an orthonormal basis
is called rectangular
(or orthonormal)
system of coordinates. If a system of coordinates
is given, then for an arbitrary point M
in space can be put in one-to-one correspondence the vector
of
which the beginning is in O
and the end – in M.
Vector
is called the radius-vector
of M
in the system of coordinates
.
The coordinates of the radius-vector of M are called the coordinates
of
M
in the system of coordinates
.
Let
the coordinates of non-coinciding points M1
and M2
in some common Cartesian system of coordinates
with
and
be given. Find the point M such that
.
Solution:
.
Since
,
we have
.
Consequently,
.
18. Complex numbers. Actions over complex numbers. Algebraic and trigonometric forms of a complex number.
Complex numbers are expressions of the following form a + bi (a and b are real numbers, i is the imaginary unit).
Two complex numbers a1 + b1i and a2 + b2i are equal а1 = а2 and b1 = b2.
The sum of numbers a1 + b1i and a2 + b2i is called the number a1 + a2 + (b1 + b2) i.
The product of numbers a1 + b1i and a2 + b2i is called the number a1a2 – b1b2 + (a1b2 + a2b1) i.
The set of all complex numbers is denoted by С. The set of real numbers R is a sunset of the set of complex numbers, i.e. R C. A real number а is the real part of a complex number a + bi. A real number b is the imaginary part of the complex number a + bi.
Numbers a + bi and a – bi, i.e. numbers differing only the sign of the imaginary part, are called conjugate complex numbers.
The
module of
a complex number z
= a + bi
is denoted by |z|
and is determined by the formula
Let's consider a plane with rectangular system of coordinates Оху. The point of a plane z (x, y) can be put to each complex number z = x + yi in correspondence, and this correspondence is one-to-one. A plane on which such correspondence is realized is called a complex plane. The real numbers z = x + 0i = x are located on the axis Ox; and therefore it is called the real axis. Pure imaginary numbers z = 0 + yi = yi are located on the axis Оу; it is called the imaginary axis.
Observe
that r
= |z|
represents the distance between the point z
and
the origin of coordinates. Every point z
is
corresponded the radius-vector
of
the point z.
The angle formed by the radius-vector of the point z
with the axis Ох
is called the argument
= Arg z
of the point. Here –
< Arg z < + .
Y
z
y
О х Х
Every
solution
of the system of equations
is called the argument of a complex number z
= x + yi
0.
All the arguments of a number z
are differed on whole multiples 2
and are denoted by one symbol Arg
z.
The value of Arg
z
satisfying the condition 0
Arg
z < 2
is called the principal
value
of the argument and is denoted by arg
z.
The formulas (*) imply that for every complex number z the following equality is true:
z = |z| (cos + i sin)
It is called the trigonometric form of the number z.
And the form of a complex number z = x + yi is called algebraic.