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Îñíîâí³ âëàñòèâîñò³ ñêàëÿðíîãî äîáóòêó òàê³:<br />
r r r r<br />
1. a⋅ b = b⋅a<br />
.<br />
r r r r r r<br />
2. λ( ab ⋅ ) = ( λa) ⋅ b= a⋅( λb)<br />
.<br />
r r r r r r r<br />
3. a⋅ ( b+ c)<br />
= a⋅ b+ a⋅c<br />
.<br />
r r r r<br />
r<br />
4. a⋅a<br />
≥0, a⋅ a = 0 òîä³ ³ ò³ëüêè òîä³, êîëè a = 0 .<br />
Âåêòîðè, ñêàëÿðíèé äîáóòîê ÿêèõ äîð³âíþº íóëþ, íàçèâàþòüñÿ<br />
îðòîãîíàëüíèìè. Ëåãêî ïîáà÷èòè, ùî âåêòîðè áàçèñó<br />
r r r<br />
r r<br />
e1, e2,..., en<br />
ïîïàðíî îðòîãîíàëüí³, òîáòî ei<br />
⋅ e j<br />
= 0 ïðè i ≠ j, à<br />
r r<br />
ei<br />
⋅ e j<br />
= 1 ïðè i = j. Äëÿ äâîâèì³ðíèõ ³ òðèâèì³ðíèõ ïðîñòîð³â<br />
ïîíÿòòÿ îðòîãîíàëüíîñò³ òà ïåðïåíäèêóëÿðíîñò³ çá³ãàþòüñÿ.<br />
ßêùî â ë³í³éíîìó ïðîñòîð³ âèçíà÷åíî ñêàëÿðíèé äîáóòîê<br />
äâîõ âåêòîð³â çà ïðàâèëîì (2.1.4) ç âëàñòèâîñòÿìè 1 – 4, òî<br />
â³í íàçèâàºòüñÿ åâêë³äîâèì.<br />
 åâêë³äîâîìó ïðîñòîð³ â³äñòàíü ì³æ äâîìà òî÷êàìè<br />
À (õ 1 , õ 2 ,…, õ n ) ³  (ó 1 , ó 2 ,…, y n ) âèçíà÷àºòüñÿ ÿê äîâæèíà<br />
ur<br />
AB = a = y1 −x1, y2 −x2,..., y n<br />
−x n<br />
, òîáòî<br />
âåêòîðà ( )<br />
uuuur r r r r2<br />
2 2 2<br />
AB = AB = a = a ⋅ a = a = y − x + y − x + + y −x .<br />
( ) ( ) ( )<br />
1 1 2 2<br />
...<br />
n n<br />
2.1.7. Ïðîñò³ð òîâàð³â, âåêòîð ö³í òà âàðò³ñòü íàáîðó<br />
òîâàð³â<br />
ßê â³äîìî, â åêîíîì³ö³ ï³ä òîâàðîì ðîçó쳺òüñÿ äåÿêå<br />
áëàãî àáî ïîñëóãà, ÿê³ íàä³éøëè â ïðîäàæ ó âèçíà÷åíèé ÷àñ<br />
³ ó âèçíà÷åíîìó ì³ñö³. Áóäåìî ââàæàòè, ùî º ï ð³çíèõ òîâàð³â.<br />
×åðåç õ ³ ïîçíà÷èìî ê³ëüê³ñòü ³-ãî òîâàðó, òîä³ íàá³ð<br />
òîâàð³â ìîæíà ïîçíà÷èòè ÷åðåç n-âèì³ðíèé âåêòîð<br />
Õ=(õ 1 , ..., õ n ). Ìíîæèíà âñ³õ íàáîð³â òîâàð³â íàçèâàºòüñÿ<br />
ïðîñòîðîì òîâàð³â. Öåé ïðîñò³ð ïîçíà÷èìî áóêâîþ Ñ. Îêðåì³<br />
âåêòîðè áóäåìî íàçèâàòè åëåìåíòàìè ïðîñòîðó Ñ.<br />
Îñê³ëüêè ê³ëüê³ñòü òîâàð³â íå ìîæå áóòè â³ä’ºìíîþ, òî î÷åâèäíî,<br />
ùî ïðîñò³ð Ñ º ï³äìíîæèíîþ ë³í³éíîãî ïðîñòîðó âåêòîð³â<br />
ðîçì³ðíîñò³ n. Ïðîñò³ð Ñ ñêëàäàºòüñÿ ³ç âåêòîð³â,<br />
êîìïîíåíòè ÿêèõ º íåâ³ä’ºìí³ ÷èñëà. Ó ïðîñòîð³ òîâàð³â Ñ<br />
ìîæíà äîäàâàòè âåêòîðè ³ ìíîæèòè ¿õ íà íåâ³ä’ºìí³ ä³éñí³<br />
÷èñëà (íà ïðàêòèö³ âåêòîðè ïîìíîæóþòüñÿ íà íåâ³ä’ºìí³<br />
ðàö³îíàëüí³ ÷èñëà).<br />
Ïðèïóñòèìî òàêîæ, ùî êîæíèé òîâàð ìຠö³íó. Î÷åâèäíî,<br />
ùî óñ³ ö³íè º ñòðîãî äîäàòíèìè.<br />
Íåõàé ö³íà îäèíèö³ ³-ãî òîâàðó º ð ³ , òîä³ âåêòîð<br />
P r =(ð 1 , ..., ð n ) º âåêòîð ö³í. Íàá³ð òîâàð³â ÿê âåêòîð ìຠòó<br />
ñàìó ðîçì³ðí³ñòü, ùî ³ âåêòîð ö³í. Äëÿ íàáîðó òîâàð³â<br />
X r =(õ 1 , ..., õ n ) ³ âåêòîðà ö³í ¿õ ñêàëÿðíèé äîáóòîê º ÷èñëî<br />
PX<br />
rr = p 1 x 1 +...+ p n x n , ÿêå íàçèâàºòüñÿ ö³íîþ íàáîðó, àáî éîãî<br />
âàðò³ñòþ, ³ ïîçíà÷àºòüñÿ ñèìâîëîì ñ ( X r ) .<br />
Ðîçãëÿíåìî òàêîæ ïîíÿòòÿ ³íòåíñèâíîñò³ ðîáîòè ï³äïðè-<br />
ºìñòâà. Ïðèïóñòèìî, ùî ï³ä âïëèâîì äåÿêèõ ôàêòîð³â (çîâí³øí³õ<br />
³ âíóòð³øí³õ) âèðîáíèöòâî êîæíîãî ç òîâàð³â çá³ëüøó-<br />
ºòüñÿ àáî çìåíøóºòüñÿ â λ ðàç³â. Äëÿ õàðàêòåðèñòèêè ïðîöåñ³â<br />
òàêîãî òèïó ââåäåìî ïàðàìåòð λ 1 , ÿêèé õàðàêòåðèçóº<br />
³íòåíñèâí³ñòü ðîáîòè ï³äïðèºìñòâà é îá÷èñëþºòüñÿ òàê:<br />
λ<br />
1<br />
= ⎨<br />
⎪λ<br />
⎧λ,<br />
ÿêùî âèðîáíèöòâî òîâàð³â â λ ðàç³â çá³ëüøóºòüñÿ,<br />
⎪<br />
1 , ÿêùî âèðîáíèöòâî òîâàð³â â λ ðàç³â çìåíøóºòüñÿ.<br />
⎩<br />
Ïðè λ 1 =1 áóäåìî ââàæàòè, ùî ³íòåíñèâí³ñòü ðîáîòè ï³äïðèºìñòâà<br />
íîðìàëüíà (çâè÷àéíà). ²íø³ ³íòåíñèâíîñò³ ïîð³âíþþòüñÿ<br />
ç íîðìàëüíîþ. Òàêèì ÷èíîì, ÿêùî ï³äïðèºìñòâî<br />
ïðè íîðìàëüí³é ³íòåíñèâíîñò³ âèðîáëÿº âåêòîð òîâàð³â<br />
X r =(õ,, ... õ n ), òî ïðè ³íòåíñèâíîñò³ λ 1 (λ 1 ≠1) áóäå âèðîáëÿòè<br />
λ 1 X r òîâàð³â.<br />
Àíàëîã³÷íî ìîæíà ââåñòè ïîíÿòòÿ ³íòåíñèâíîñò³ ïðîäàæó<br />
òîâàð³â.<br />
Ïðèêëàä 2.1.4. ʳîñê çà äåíü ïðîäຠãàçåòè 50 íàéìåíóâàíü.<br />
Íàâåä³òü ìîæëèâ³ âàð³àíòè ââåäåííÿ åëåìåíò³â ïðîñòîðó<br />
òîâàð³â.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Îáìåæèìîñÿ ò³ëüêè äâîìà âàð³àíòàìè:<br />
1 0 . Ó ÿêîñò³ åëåìåíòà ïðîñòîðó Ñ ìîæíà âçÿòè âåêòîð<br />
ðîçì³ðíîñò³ 50, äå êîìïîíåíòè âåêòîðà ÿâëÿþòü ñîáîþ ê³ëüê³ñòü<br />
ïðîäàíèõ ãàçåò ïî 50 íàéìåíóâàííÿì êîæíî¿.<br />
38 39