Vector Product and its Properties
We have considered a product of two vectors equal to a number (inner product)
.
What happens if the product of two vectors is a vector:
?
C
onsider
two vectors
and
:
Definition 2. The
vector product of two vectors
and
is a vector
,
satisfying the following conditions:
(1) the absolute value of equals the product of the absolute values of the two given vectors and the sine of the angle between them:
;
(*)
(2) the vector
is perpendicular to both vectors
and
:
;
(3) the three vectors , and constitute a right triple of vectors (that is, looking from the tail of, we see that the shorter rotation from to is carried out anticlockwise). The vector product of and is denoted by
.
Property 1. The absolute value of the vector product of two vectors is equal to the area of the parallelogram spanned by these vectors:
.
Property 2. The vector product is anticommutative, i.e.,
.
Property 3. To multiply a vector product by a number , it is suffices to multiply one of the vectors by this number (without proof):
.
Property 4. Vector product is associative:
.
Property 5. The vector product of collinear vectors is equal to zero, and vice versa, if the vector product of two vectors is zero, then these vectors are collinear.
The vector product in coordinates. Consider vectors
and
.
Let us find the coordinate of their vector product. Property 4 allows us to multiply them term by term
.
Since the vector products of collinear vectors equal zero, it follows that the first, fifth and ninth terms are null vectors:
Consider
the vector products of the unit mutually perpendicular vectors
and
.
Since,
,
and
is a right triple of vectors, it
follows from the definition of vector product and its commutativity
that
;
;
;
;
;
.
(*)
Substituting the products of unit vectors into the required vector product, we obtain
Note that the
right – hand side is the expansion of a third – order determinant
along the row with elements
and
.
Thus, the coordinates of the vector product are determined form the third – order determinant as
,
and its absolute value is
.
Triple Product of Vectors and its Properties
Given
vectors
and
,
we could take the inner product
and multiply it by the third vector
,
thus obtaining
,
which is collinear to
.
This is uninteresting.
Thus, we consider the vector product
and take its products with vector
of two kinds:
(1) inner and
(2) vector.
What we obtain is called the scalar triple product and the double vector product of and .
Definition 3. The
triple product of three vectors is the
inner product of the third vector by the vector product of the first
two vectors; it is denoted by
.
Definition 4.
The vector product of the vector product of the first two vectors and
the third vector ones is called the double vector product:
.
Since double vector product is used very rarely, it have been little studied.
Suggestion for reflection. Students are suggested to study, investigate, and apply the double vector product.
In what follows, we consider only scalar triple product.
Property 1. The triple product of three vectors equals the volume of a parallelepiped spanned by these three vectors.
Property 2. Triple
product is commutative, and
.
The proof is left to the reader.
Property 3. A constant multiplier of any vector can be factored out of scalar triple product:
.