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
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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
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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:
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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
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defined in an orthonormal basis, is calculated by the formula
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The cosine of the angle between two non-zero vectors
and
may be calculated by the formula
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where and
.
The cosines of the angles formed by the vector with the basis vectors
are said to be the direction cosines of
:
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The direction cosines have the following property:
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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:
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and homogeneous:
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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
:
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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
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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
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The mixed product of three vectors
is the scalar product of
and the vector product of the vectors
and
:
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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:
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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=42733