 |
АвтоАвтоматизацияАрхитектураАстрономияАудитБиологияБухгалтерияВоенное делоГенетикаГеографияГеологияГосударствоДомДругоеЖурналистика и СМИИзобретательствоИностранные языкиИнформатикаИскусствоИсторияКомпьютерыКулинарияКультураЛексикологияЛитератураЛогикаМаркетингМатематикаМашиностроениеМедицинаМенеджментМеталлы и СваркаМеханикаМузыкаНаселениеОбразованиеОхрана безопасности жизниОхрана ТрудаПедагогикаПолитикаПравоПриборостроениеПрограммированиеПроизводствоПромышленностьПсихологияРадиоРегилияСвязьСоциологияСпортСтандартизацияСтроительствоТехнологииТорговляТуризмФизикаФизиологияФилософияФинансыХимияХозяйствоЦеннообразованиеЧерчениеЭкологияЭконометрикаЭкономикаЭлектроникаЮриспунденкция
Inner Product and its Properties
Definition 1. The inner product of two vectors 
and 
is the product of the absolute values of these vectors and the cosine of the angle between them:
.
Property 1. Theinner product of two vectors is equal to the product of the absolute value of one vector and the projection of the second vector onto the first, i.e.,
.
Property 2. The inner product of two vectors is equal to zero if and only if these vectors are perpendicular.
1. If 
, then 
, and the inner product equals 
.
2. Conversely, suppose that 
Property 3. The inner product of vectors is commutative:
Property 4. To multiply an inner product by a number l, it is sufficient to multiply one of the factors by l:
(Without proof.)
Property 5. Inner product is associative:
.
and 
.
The last property allows us to multiply these vectors term by term:
.
Since the unit vectors are mutually perpendicular (
), their inner products are zero (by Property 2).
Consider the product of a unit vector with itself:
; 
; 
.
Thus, six of the nine terms are zero, and the remaining three terms are
.
The inner product of vectors is equal to the sum of products of their coordinates.
Поиск по сайту: