Vector algebra
2020 Mathematics Subject Classification: Primary: 15A72 [MSN][ZBL]
A branch of vector calculus dealing with the simplest operations involving (free) vectors (cf. Vector). These include linear operations, viz. addition of vectors and multiplication of a vector by a number.
The sum 
of two vectors 
and 
is the vector drawn from the origin of 
to the end of 
if the end of 
and the origin of 
coincide. The operation of vector addition has the following properties:
(commutativity);
(associativity);
(existence of a zero-element);
(existence of an inverse element).
Here 
is the zero vector, and 
is the vector opposite to the vector 
(its inverse). The difference 
of two vectors 
and 
is the vector 
for which 
.
The product 
of a vector 
by a number 
is, if 
, 
, the vector whose modulus equals 
and whose direction is that of 
if 
, and that of the inverse of 
if 
. If 
or (and) 
, then 
. The operation of multiplication of a vector by a number has the properties:
(distributivity with respect to vector addition);
(distributivity with respect to addition of numbers);
(associativity);
(multiplication by one).
The set of all free vectors of a space with the induced operations of addition and multiplication by a number forms a vector space (a linear space). Below "vector" means free vector, or equivalently, element of a given vector space.
An important concept in vector algebra is that of linear dependence of vectors. Vectors 
are said to be linearly dependent if there exist numbers 
, at least one of which is non-zero, such that the equation
 | (1) |
is valid. For two vectors to be linearly dependent it is necessary and sufficient that they are collinear; for three vectors to be linearly dependent it is necessary and sufficient that they are coplanar. If one of the vectors 
is zero, the vectors are linearly dependent. The vectors 
are said to be linearly independent if it follows from (1) that the numbers 
are equal to zero. At most two, respectively three, linearly independent vectors exist in a plane, respectively three-dimensional space.
A set of three (two) linearly independent vectors 
of three-dimensional space (a plane), taken in a certain order, forms a basis. Any vector 
can be uniquely represented as the sum
 |
The numbers 
are said to be the coordinates (components) of 
in the given basis; this is written as 
.
Two vectors 
and 
are equal if and only if their coordinates in the same basis are equal. A necessary and sufficient condition for two vectors 
and 
, 
, to be collinear is proportionality of their corresponding coordinates: 
, 
, 
. A necessary and sufficient condition for three vectors 
, 
and 
to be coplanar is the equality
 |
Linear operations on vectors can be reduced to linear operations on coordinates. The coordinates of the sum of two vectors 
and 
are equal to the sums of the corresponding coordinates: 
. The coordinates of the product of the vector 
by a number 
are equal to the products of the coordinates of 
by 
: 
.
The scalar product (or inner product) 
of two non-zero vectors 
and 
is the product of their moduli by the cosine of the angle 
between them:
 |
In this context, 
is understood as the angle between the vectors that does not exceeding 
. If 
or 
, their scalar product is defined as zero. The scalar product has the following properties:
(commutativity);
(distributivity with respect to vector addition);
(associativity with respect to multiplication by a number);
only if 
and/or 
, or 
.
Scalar vector products are often calculated using orthogonal Cartesian coordinates, i.e. vector coordinates in a basis consisting of mutually perpendicular unit vectors 
(an orthonormal basis). The scalar product of two vectors
 |
defined in an orthonormal basis, is calculated by the formula
 |
The cosine of the angle 
between two non-zero vectors 
and 
may be calculated by the formula
 |
where 
and 
.
The cosines of the angles formed by the vector 
with the basis vectors 
are said to be the direction cosines of 
:
 |
 |
The direction cosines have the following property:
 |
A straight line with a unit vector 
chosen on it, which specifies the positive direction on the straight line, is said to be an axis. The projection 
of a vector 
onto the axis is the directed segment on the axis whose algebraic value is equal to the scalar product of 
and 
. Projections are additive:
 |
and homogeneous:
 |
Each coordinate of a vector in an orthonormal basis is equal to the projection of this vector on the axis defined by the respective basis vector.
Figure: v096350a
Left and right vector triples are distinguished in space. A triple of non-coplanar vectors 
is said to be right if, to the observer at the common vector origin, the movement 
, in that order, appears to be clockwise. If it appears to be counterclockwise, 
is a left triple. The direction in space of the right (left) vector triples may be represented by stretching out the thumb, index finger and middle finger of the right (left) hand, as shown in the figure. All right (left) vector triples are said to be identically directed. In what follows, the vector triple of basis vectors 
will be assumed to be a right triple.
Let the direction of positive rotation (from 
to 
) be given on a plane. Then the pseudo-scalar product 
of two non-zero vectors 
and 
is defined as the product of their lengths (moduli) by the sine of the angle 
of positive rotation from 
to 
:
 |
By definition, if 
or 
is zero, their pseudo-scalar product is set equal to zero. The pseudo-scalar product has the following properties:
(anti-commutativity);
(distributivity with respect to vector addition);
(associativity with respect to multiplication by a number);
only if 
and/or 
, or if 
and 
are collinear.
If, in an orthonormal basis, the vectors 
and 
have coordinates 
and 
, then
 |
The vector product 
of two non-zero non-collinear vectors 
and 
is the vector whose modulus is equal to the product of the moduli by the sine of the angle 
between them, which is perpendicular to 
and to 
and is so directed that the vector triple 
is a right triple:
 |
This product is defined as zero if 
and/or 
, or if the two vectors are collinear. The vector product has the following properties:
(anti-commutativity);
(distributivity with respect to vector addition);
(associativity with respect to multiplication by a number);
only if 
and/or 
, or if 
and 
are collinear.
If the coordinates of two vectors 
and 
in an orthonormal basis are 
and 
, then
 |
The mixed product 
of three vectors 
is the scalar product of 
and the vector product of the vectors 
and 
:
 |
The mixed product has the following properties:
 |
 |
only if 
and/or 
and/or 
, or if the vectors 
are coplanar;
if the vector triple 
is a right triple; 
if 
is a left triple.
The modulus of the mixed product is equal to the volume of the parallelepipedon constructed on the vectors 
. If, in an orthonormal basis, the vectors 
, 
and 
have coordinates 
, 
and 
, then
 |
The double vector product 
of three vectors 
is 
.
The following formulas are used in calculating double vector products:
 |
 |
 |
 |
References
| [1] | P.S. Aleksandrov, "Lectures on analytical geometry" , Moscow (1968) (In Russian) |
| [2] | N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian) |
| [3] | V.A. Il'in, E.G. Poznyak, "Analytical geometry" , MIR (1984) (Translated from Russian) |
| [4] | A.V. Pogorelov, "Analytical geometry" , Moscow (1968) (In Russian) |
Comments
References
| [a1] | P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958) |
| [a2] | R. Capildeo, "Vector algebra and mechanics" , Addison-Wesley (1968) |
Vector algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_algebra&oldid=49132