Principles of vector calculus
Vectors. Opposite vectors. Zero vector. Length (modulus) of vector.
Collinear vectors. Coplanar vectors. Equality of vectors. Parallel
transfer of vectors. Addition of vectors. Subtraction of vectors.
Laws of addition of vectors. Laws of multiplication of vector by a
number. Scalar product of vectors. Angle between non-zero vectors.
Scalar square. Properties of a scalar product. Unit orthogonal vectors.
Coordinates of a vector. Algebraic operations with vectors. Vector
product of vectors. Properties of a vector product. Necessary and
sufficient condition of collinearity of vectors. Necessary and sufficient
condition of coplanarity of vectors.
Vector is a directed segment, connecting two
points in a space ( in a plane ). Vectors are signed usually either
by small letters or by initial and final points. For example, a vector directed from
point A to point B can be signed
as
a,
__
A zero vector 0 or 0 is a
vector, for which initial and final points coincide,
i.e. A=B. From here it follows: 0 = – 0.
A length (
modulus
) of vector
a, signed as | a |, is a length of its imaging segment AB. Particularly, | 0
| = 0.
Vectors are called collinear
ones,
if their directed segments belong to parallel lines.
Collinear vectors a and b are signed as a || b
.
Three or more vectors are called
coplanar, if they lie in the same plane.
Equality of vectors.
Two vectors a and b are equal, if they are collinear ones and their lengths are equal,
i.e. a
|| b and | a | = | b | . Hence, vectors don’t change at
a parallel transfer.
Addition of vectors.
As vectors are directed segments,
then their addition can be executed geometrically. (Algebraic
addition of vectors see below, in the point “Unit orthogonal vectors”).
Assume, that
__ __
a = AB and b
= CD ,
then __
__
a + b = AB + CD
is a vector, received after executing
of the two operations:
a) a parallel transfer of one of the
vectors till its initial point will coincide with
a final point of another vector;
b) a geometrical addition by drawing
the resulting vector from an initial point
of an immovable vector to a final
point of a transferred vector.
Subtraction of vectors.
This operation is reduced to the
previous by changing a subtracted vector to an opposite
one:
a
– b =
a + ( – b ) .
Laws of addition.
I. a + b = b
+ a
( C o m m u t a t i v i t y ).
II.
( a
+ b ) + c
= a
+ ( b
+ c
) ( A s s o c i a t i v i t y ).
III.
a
+ 0 = a .
IV.
a
+ (– a ) = 0 .
Laws of multiplication of vector
by a number.
I. 1 · a = a ,
0 · a = 0 , m · 0
= 0 , ( –1 ) · a = – a .
II. m a = a m , | m a | = | m | · | a | .
III. m ( n a
) = ( m n )
a . ( A s s o c i a t i v i t y of
multiplication
by a number ).
IV.
( m + n ) a = m a + n a , ( D i s t r i b u t i v i t y of
m
( a + b ) = m a + m b . multiplication by a number ).
__ __
Scalar product of vectors.
An
angle between non-zero vectors AB and CD is an angle, formed at a parallel transfer
one of the vectors till coinciding the points A and C. A scalar
product of vectors a and
b is called a number,
equal to a product of lengths (
modules )
of these vectors by cosine of angle between them:
If one of vectors is a zero vector,
then a scalar product of these vectors is equal to zero by the definition:
( a , 0 ) = ( 0 , b ) = 0 .
If both vectors are non-zero ones,
then cosine of the angle between them may be found by the formula:
A scalar product (
a , a ), equal to | a
| ² ,
is called a scalar square.
A length of vector
a and its scalar square are tied by the
relation:
A scalar product of two vectors
is:
- positive, if an angle
between the vectors is acute ;
- negative, if an angle
between the vectors is obtuse .
A scalar product of two non-zero
vectors is equal to
zero, if and only if an angle between the
vectors is right, i.e. these vectors are perpendicular ( orthogonal ):
Properties of a scalar product.
For any vectors
a , b , c and any number m the following relations are
valid:
I.
( a , b ) = ( b , a ) . ( C
o m m u t a t i v i t y )
II. (
m a , b ) = m (
a , b
) .
III.
( a + b , c ) = ( a , c ) + ( b , c ). ( D i s t r i b u t i v i t
y )
Unit orthogonal vectors.
In any
rectangular system of coordinates it is possible to introduce unit two-and-two
orthogonal vectors i, j and k, connected with coordinate axes:
i – for x-axis, j – for y-axis and
k – for z-axis. According to this definition we have:
( i , j ) = ( i
, k
) = ( j
, k
) = 0,
| i | =
| j | = | k | =
1.
Any vector
a can be expressed through these vectors
by the only way: a
= x i + y j + z k
. Another form of the record is: a = ( x, y, z ) . Here x, y, z – coordinates
of the vector
a in this system of coordinates. According to the last relation
and properties of the unit orthogonal vectors i , j , k a scalar product of two vectors can be
written in another shape. Assume a = ( x, y, z ); b
= ( u, v, w ). Then ( a , b ) = xu + yv + zw. A scalar product of two vectors is equal to a sum of
products of corresponding
coordinates.
A length (modulus) of vector
a = ( x,
y, z
) is
equal to:
Besides, we receive now the possibility
for algebraic operations with vectors; namely, addition and subtraction of
vectors can be executed by coordinates:
a + b = ( x + u , y + v ,
z + w ) ;
a –
b = ( x
–
u , y –
v , z –
w ) .
Vector product of vectors.
A vector product [
a , b
] of vectors a and b ( in the indicated order ) is a vector:
There is another formula for a length of vector [
a, b
]
:
/\
| [ a, b ] | = | a | | b
| sin (
a, b )
,
i.e. a
length ( modulus ) of vector product of the vectors
a and b is equal to product of lengths (
modules )
of these vectors by sine of the angle between them.
Differently this fact can be interpreted
as following : a length ( modulus ) of vector
[ a, b
] is equal numerically to an area of parallelogram,
built on vectors a and b .
Properties of vector product.
I. A vector
[ a, b ]
is perpendicular (orthogonal) both to vector
a and vector b. ( Prove this, please ! ) .
II. [
a , b ] = – [ b , a ] . ( A n t i c o m m u t a t i v
i t y ).
III. [ m a , b ] = m [ a , b ] .
IV. [ a + b , c ] = [ a , c ] + [
b , c ] . ( D i s t r i b u t i v
i t y ).
V. [
a , [ b , c ] ] = b ( a , c ) – c (
a , b ) .
VI. [ [ a , b ] , c ] = b ( a , c ) – a ( b , c ) .
Necessary
and sufficient condition of collinearity of vectors a
= ( x, y, z ) and
b = ( u,
v, w ) :
Necessary and sufficient condition of coplanarity of vectors a
= ( x, y, z ) , b = ( u, v, w ) and c
= ( p, q, r ) :
E x a m p l e . The vectors: a
= ( 1, 2, 3 ) and b = ( – 2 , 0 ,4 ) are given. Calculate
their scalar product, vector product and an angle between these vectors.
S o l u t i o n . Using the
corresponding formulas (see above), we’ll receive:
a). the scalar product: (
a , b ) = 1
· ( – 2 ) + 2 ·
0 + 3 · 4 = 10 ;
b). the vector product:
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