 |
АвтоАвтоматизацияАрхитектураАстрономияАудитБиологияБухгалтерияВоенное делоГенетикаГеографияГеологияГосударствоДомДругоеЖурналистика и СМИИзобретательствоИностранные языкиИнформатикаИскусствоИсторияКомпьютерыКулинарияКультураЛексикологияЛитератураЛогикаМаркетингМатематикаМашиностроениеМедицинаМенеджментМеталлы и СваркаМеханикаМузыкаНаселениеОбразованиеОхрана безопасности жизниОхрана ТрудаПедагогикаПолитикаПравоПриборостроениеПрограммированиеПроизводствоПромышленностьПсихологияРадиоРегилияСвязьСоциологияСпортСтандартизацияСтроительствоТехнологииТорговляТуризмФизикаФизиологияФилософияФинансыХимияХозяйствоЦеннообразованиеЧерчениеЭкологияЭконометрикаЭкономикаЭлектроникаЮриспунденкция
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. Thetriple 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:
.
Triple Product in Coordinates. Given three vectors 
, 
, and 
, let us express the triple product of these vectors in terms of their coordinates. Consider the triple product
.
The vector product equals
.
Taking its inner product with , we obtain
;
this is a third – order determinant expanded along the last line, i.e.,
.
Thus, the triple product of three vectors equals the third – order determinant of the composed of the coordinates of these vectors.
Поиск по сайту: