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Векторски производ на два вектора

Module by: Liljana Stefanovska. E-mail the author

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Summary: Се дефинира векторски производ на два вектора и неговите својства. Cross product and properties

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ВЕКТОРСКИ ПРОИЗВОД НА ДВА ВЕКТОРА

За решавање на проблеми во кои се бара да се определи вектор кој ќе биде нормален на два дадени вектора, се дефинира друг вид на производ кој е векторска величина и се нарекува векторски производ и дефинира со:

Дефиницијa. Векторски производ на векторите a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {} и b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {} е вектор c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {} кој се означува со

c → = a → × b → c → = a → × b → size 12{ {c} cSup { size 8{ rightarrow } } = {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}

и е определен со:

- должина | c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {}| = | a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}|| b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}| sin ∠∠ size 12{∠} {}( a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}, b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}) ;

- правец кој е нормален на рамнината во која лежат векторите a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {} и b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}, односно

c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {}⊥ a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {} и c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {}⊥ b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {};

- насоката на c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {}е таква што векторите a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}, b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {} и c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {} образуваат десна тројка вектори, односно се однесуваат по правилото на десен винт (Сл. 1.8.).

Table 1
graphics1.jpg
Слика 1.8. Правило на десен винт

Својства на векторскиот производ

Од дефиницијата за векторски производ следуваат следните негови својства:

1. a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}= −b→×a→−b→×a→ size 12{ - {b} cSup { size 8{ rightarrow } } times {a} cSup { size 8{ rightarrow } } } {} (антикомутативен закон)

2. a→×(b→+c→)=a→×b→+a→×c→a→×(b→+c→)=a→×b→+a→×c→ size 12{ {a} cSup { size 8{ rightarrow } } times \( {b} cSup { size 8{ rightarrow } } + {c} cSup { size 8{ rightarrow } } \) = {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } + {a} cSup { size 8{ rightarrow } } times {c} cSup { size 8{ rightarrow } } } {} (дистрибутивен закон)

3. λ( a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}) = (λ a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {})× b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}= a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}× (λ b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}) (множење со скалар λ)

4. Ако a→≠0→a→≠0→ size 12{ {a} cSup { size 8{ rightarrow } } <> {0} cSup { size 8{ rightarrow } } } {} и b→≠0→b→≠0→ size 12{ {b} cSup { size 8{ rightarrow } } <> {0} cSup { size 8{ rightarrow } } } {} тогаш,

a→×b→=0→a→×b→=0→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } = {0} cSup { size 8{ rightarrow } } } {}⇔a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {} и b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {} се колинеарни вектори ( a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}|| b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}).

5. Векторскиот производ меѓу единичните вектори е:

i→×j→=k→i→×j→=k→ size 12{ {i} cSup { size 8{ rightarrow } } times {j} cSup { size 8{ rightarrow } } = {k} cSup { size 8{ rightarrow } } } {}, j→×k→=i→j→×k→=i→ size 12{ {j} cSup { size 8{ rightarrow } } times {k} cSup { size 8{ rightarrow } } = {i} cSup { size 8{ rightarrow } } } {}, k→×i→=j→k→×i→=j→ size 12{ {k} cSup { size 8{ rightarrow } } times {i} cSup { size 8{ rightarrow } } = {j} cSup { size 8{ rightarrow } } } {},

i→×i→=0→i→×i→=0→ size 12{ {i} cSup { size 8{ rightarrow } } times {i} cSup { size 8{ rightarrow } } = {0} cSup { size 8{ rightarrow } } } {}, j→×j→=0→j→×j→=0→ size 12{ {j} cSup { size 8{ rightarrow } } times {j} cSup { size 8{ rightarrow } } = {0} cSup { size 8{ rightarrow } } } {}, k→×k→=0→k→×k→=0→ size 12{ {k} cSup { size 8{ rightarrow } } times {k} cSup { size 8{ rightarrow } } = {0} cSup { size 8{ rightarrow } } } {}.

Ако векторите a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {} и b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {} се зададени со своите координати

a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}= {x1, y1, z1} и b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}= {x2, y2, z2},

нивниот векторски производ изразен преку координати е

a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}= (x1i→+y1j→+z1k→)×(x2i→+y2j→+z2k→)=(x1i→+y1j→+z1k→)×(x2i→+y2j→+z2k→)= size 12{ \( x rSub { size 8{1} } {i} cSup { size 8{ rightarrow } } +y rSub { size 8{1} } {j} cSup { size 8{ rightarrow } } +z rSub { size 8{1} } {k} cSup { size 8{ rightarrow } } \) times \( x rSub { size 8{2} } {i} cSup { size 8{ rightarrow } } +y rSub { size 8{2} } {j} cSup { size 8{ rightarrow } } +z rSub { size 8{2} } {k} cSup { size 8{ rightarrow } } \) ={}} {}

= x 1 x 2 ( i → × i → ) + x 1 y 2 ( i → × j → ) + x 1 z 2 ( i → × k ) + + y 1 x 2 ( j → × i → ) + y 1 y 2 ( j → × j → ) + y 1 z 2 ( j → × k ) + + z 1 x 2 ( k → × i → ) + z 1 y 2 ( k → × j → ) + z 1 z 2 ( k → × k → ) = = x 1 x 2 ( i → × i → ) + x 1 y 2 ( i → × j → ) + x 1 z 2 ( i → × k ) + + y 1 x 2 ( j → × i → ) + y 1 y 2 ( j → × j → ) + y 1 z 2 ( j → × k ) + + z 1 x 2 ( k → × i → ) + z 1 y 2 ( k → × j → ) + z 1 z 2 ( k → × k → ) = alignl { stack { size 12{ {}=x rSub { size 8{1} } x rSub { size 8{2} } \( {i} cSup { size 8{ rightarrow } } times {i} cSup { size 8{ rightarrow } } \) +x rSub { size 8{1} } y rSub { size 8{2} } \( {i} cSup { size 8{ rightarrow } } times {j} cSup { size 8{ rightarrow } } \) +x rSub { size 8{1} } z rSub { size 8{2} } \( {i} cSup { size 8{ rightarrow } } times k \) +{}} {} # +y rSub { size 8{1} } x rSub { size 8{2} } \( {j} cSup { size 8{ rightarrow } } times {i} cSup { size 8{ rightarrow } } \) +y rSub { size 8{1} } y rSub { size 8{2} } \( {j} cSup { size 8{ rightarrow } } times {j} cSup { size 8{ rightarrow } } \) +y rSub { size 8{1} } z rSub { size 8{2} } \( {j} cSup { size 8{ rightarrow } } times k \) +{} {} # +z rSub { size 8{1} } x rSub { size 8{2} } \( {k} cSup { size 8{ rightarrow } } times {i} cSup { size 8{ rightarrow } } \) +z rSub { size 8{1} } y rSub { size 8{2} } \( {k} cSup { size 8{ rightarrow } } times {j} cSup { size 8{ rightarrow } } \) +z rSub { size 8{1} } z rSub { size 8{2} } \( {k} cSup { size 8{ rightarrow } } times {k} cSup { size 8{ rightarrow } } \) ={} {} } } {}

= x 1 y 2 k → + x 1 z 2 ( − j → ) + y 1 x 2 ( − k → ) + y 1 z 2 i → + z 1 x 2 j → + z 1 y 2 ( − i → ) = = x 1 y 2 k → + x 1 z 2 ( − j → ) + y 1 x 2 ( − k → ) + y 1 z 2 i → + z 1 x 2 j → + z 1 y 2 ( − i → ) = size 12{ {}=x rSub { size 8{1} } y rSub { size 8{2} } {k} cSup { size 8{ rightarrow } } +x rSub { size 8{1} } z rSub { size 8{2} } \( - {j} cSup { size 8{ rightarrow } } \) +y rSub { size 8{1} } x rSub { size 8{2} } \( { - k} cSup { size 8{ rightarrow } } \) +y rSub { size 8{1} } z rSub { size 8{2} } {i} cSup { size 8{ rightarrow } } +z rSub { size 8{1} } x rSub { size 8{2} } {j} cSup { size 8{ rightarrow } } +z rSub { size 8{1} } y rSub { size 8{2} } \( - {i} cSup { size 8{ rightarrow } } \) ={}} {}

= ( y 1 z 2 − z 1 y 2 ) i → + ( z 1 x 2 − x 1 z 2 ) j → + ( x 1 y 2 − y 1 x 2 ) k → = = ( y 1 z 2 − z 1 y 2 ) i → + ( z 1 x 2 − x 1 z 2 ) j → + ( x 1 y 2 − y 1 x 2 ) k → = size 12{ {}= \( y rSub { size 8{1} } z rSub { size 8{2} } - z rSub { size 8{1} } y rSub { size 8{2} } \) {i} cSup { size 8{ rightarrow } } + \( z rSub { size 8{1} } x rSub { size 8{2} } - x rSub { size 8{1} } z rSub { size 8{2} } \) {j} cSup { size 8{ rightarrow } } + \( x rSub { size 8{1} } y rSub { size 8{2} } - y rSub { size 8{1} } x rSub { size 8{2} } \) {k} cSup { size 8{ rightarrow } } ={}} {}

=∣y1z1y2z2∣i→+∣z1x1z2x2∣j→+∣x1y1x2y2∣k→=∣y1z1y2z2∣i→+∣z1x1z2x2∣j→+∣x1y1x2y2∣k→ size 12{ {}= lline matrix { y rSub { size 8{1} } {} # z rSub { size 8{1} } {} ## y rSub { size 8{2} } {} # z rSub { size 8{2} } {} } rline {i} cSup { size 8{ rightarrow } } + lline matrix { z rSub { size 8{1} } {} # x rSub { size 8{1} } {} ## z rSub { size 8{2} } {} # x rSub { size 8{2} } {} } rline {j} cSup { size 8{ rightarrow } } + lline matrix { x rSub { size 8{1} } {} # y rSub { size 8{1} } {} ## x rSub { size 8{2} } {} # y rSub { size 8{2} } {} } rline {k} cSup { size 8{ rightarrow } } " "} {},

односно

a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}=∣y1z1y2z2∣, ∣z1x1z2x2∣, ∣x1y1x2y2∣=∣y1z1y2z2∣, ∣z1x1z2x2∣, ∣x1y1x2y2∣ size 12{ {}= left lbrace lline matrix { y rSub { size 8{1} } {} # z rSub { size 8{1} } {} ## y rSub { size 8{2} } {} # z rSub { size 8{2} } {} } rline ", " lline matrix { z rSub { size 8{1} } {} # x rSub { size 8{1} } {} ## z rSub { size 8{2} } {} # x rSub { size 8{2} } {} } rline ", " lline matrix { x rSub { size 8{1} } {} # y rSub { size 8{1} } {} ## x rSub { size 8{2} } {} # y rSub { size 8{2} } {} } rline right rbrace " "} {},

или изразен преку троредна детермината

a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}= ∣i→j→k→x1y1z1x2y2z2∣∣i→j→k→x1y1z1x2y2z2∣ size 12{ lline matrix { {i} cSup { size 8{ rightarrow } } {} # {j} cSup { size 8{ rightarrow } } {} # {k} cSup { size 8{ rightarrow } } {} ## x rSub { size 8{1} } {} # y rSub { size 8{1} } {} # z rSub { size 8{1} } {} ## x rSub { size 8{2} } {} # y rSub { size 8{2} } {} # z rSub { size 8{2} } {} } rline } {}.

6. Плоштината на паралелограмот образуван од векторите a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {} и b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {} е еднаква со интензитетот на нивниот векторски производ

Pпаралеограм= | a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}|,

бидејќи висината на паралелограмот е h = | b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}| sin ∠∠ size 12{∠} {}( a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}, b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}), (Сл. 1.9) .

Table 2
graphics2.jpg
Слика 1.9. Висина во паралелограм

7. Плоштината P на триаголникот чии темиња се во три точки со зададени координати A(x1, y1, z1), B(x2, y2, z2) и C(x3, y3, z3) е

P = 12∣AB→×AC∣→12∣AB→×AC∣→ size 12{ { {1} over {2} } { \lline ital "AB"} cSup { size 8{ rightarrow } } times { ital "AC" \lline } cSup { size 8{ rightarrow } } } {}.

Забелешка: Двата вектора кои го образуваат паралелограмот (триаголникот) треба да имаат почеток во иста точка.

Пример 1. Ако a→a→ size 12{ {a} cSup { size 8{ rightarrow } } } {}= {2, －1, 3}, b→b→ size 12{ {b} cSup { size 8{ rightarrow } } } {}= {0, 1, 7} и c→c→ size 12{ {c} cSup { size 8{ rightarrow } } } {}= {1, 4, 5}, да се најде

(b→×c→)×(a→×b→)(b→×c→)×(a→×b→) size 12{ \( {b} cSup { size 8{ rightarrow } } times {c} cSup { size 8{ rightarrow } } \) times \( {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } \) } {}?

Решение.Најпрво се определуваат координатите на векторите b→×c→b→×c→ size 12{ {b} cSup { size 8{ rightarrow } } times {c} cSup { size 8{ rightarrow } } } {} и a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}.

b→×c→b→×c→ size 12{ {b} cSup { size 8{ rightarrow } } times {c} cSup { size 8{ rightarrow } } } {}= ∣i→j→k→017145∣=5i→+7j→−k→−28i→=−23i→+7j→−k→∣i→j→k→017145∣=5i→+7j→−k→−28i→=−23i→+7j→−k→ size 12{ lline ` matrix { {i} cSup { size 8{ rightarrow } } {} # {j} cSup { size 8{ rightarrow } } {} # {k} cSup { size 8{ rightarrow } } {} ## 0 {} # 1 {} # 7 {} ## 1 {} # 4 {} # 5{} } ` rline `=5 {i} cSup { size 8{ rightarrow } } +7 {j} cSup { size 8{ rightarrow } } - {k} cSup { size 8{ rightarrow } } - "28" {i} cSup { size 8{ rightarrow } } = - "23" {i} cSup { size 8{ rightarrow } } +7 {j} cSup { size 8{ rightarrow } } - {k} cSup { size 8{ rightarrow } } } {}= {－23, 7, －1},

a→×b→a→×b→ size 12{ {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } } {}= ∣i→j→k→2−13017∣=−7i→+2k→−14j→−3i→=−10i→−14j→+2k→={−10,−14,2}∣i→j→k→2−13017∣=−7i→+2k→−14j→−3i→=−10i→−14j→+2k→={−10,−14,2} size 12{ = lline ` matrix { {i} cSup { size 8{ rightarrow } } {} # {j} cSup { size 8{ rightarrow } } {} # {k} cSup { size 8{ rightarrow } } {} ## 2 {} # - 1 {} # 3 {} ## 0 {} # 1 {} # 7{} } ` rline = - 7 {i} cSup { size 8{ rightarrow } } +2 {k} cSup { size 8{ rightarrow } } - "14" {j} cSup { size 8{ rightarrow } } - 3 {i} cSup { size 8{ rightarrow } } = - "10" {i} cSup { size 8{ rightarrow } } - "14" {j} cSup { size 8{ rightarrow } } +2 {k} cSup { size 8{ rightarrow } } = lbrace - "10", - "14",2 rbrace } {}.

Бараниот производ е вектор чии координати се пресметуваат со детерминантата

(b→×c→)×(a→×b→)(b→×c→)×(a→×b→) size 12{ \( {b} cSup { size 8{ rightarrow } } times {c} cSup { size 8{ rightarrow } } \) times \( {a} cSup { size 8{ rightarrow } } times {b} cSup { size 8{ rightarrow } } \) } {} = ∣i→j→k→−237−1−10−142∣=2∣i→j→k→−237−1−5−71∣=∣i→j→k→−237−1−10−142∣=2∣i→j→k→−237−1−5−71∣= size 12{ lline ` matrix { {i} cSup { size 8{ rightarrow } } {} # {j} cSup { size 8{ rightarrow } } {} # {k} cSup { size 8{ rightarrow } } {} ## - "23" {} # 7 {} # - 1 {} ## - "10" {} # - "14" {} # 2{} } ` rline `=2 lline ` matrix { {i} cSup { size 8{ rightarrow } } {} # {j} cSup { size 8{ rightarrow } } {} # {k} cSup { size 8{ rightarrow } } {} ## - "23" {} # 7 {} # - 1 {} ## - 5 {} # - 7 {} # 1{} } ` rline ={}} {}

= 2( 7i→+161k→+5j→+35k→+23j→−7i→)=2(0i→+28j→+196k→)={0,56,392}.7i→+161k→+5j→+35k→+23j→−7i→)=2(0i→+28j→+196k→)={0,56,392}. size 12{7 {i} cSup { size 8{ rightarrow } } +"161" {k} cSup { size 8{ rightarrow } } +5 {j} cSup { size 8{ rightarrow } } +"35" {k} cSup { size 8{ rightarrow } } +"23" {j} cSup { size 8{ rightarrow } } - 7 {i} cSup { size 8{ rightarrow } } \) =2 \( 0 {i} cSup { size 8{ rightarrow } } +"28" {j} cSup { size 8{ rightarrow } } +"196" {k} cSup { size 8{ rightarrow } } \) = lbrace 0,"56","392" rbrace "." } {} ◄

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