{"id":2323,"date":"2025-05-15T20:23:46","date_gmt":"2025-05-15T17:23:46","guid":{"rendered":"https:\/\/cedra.academy\/?p=2323"},"modified":"2025-09-09T20:36:02","modified_gmt":"2025-09-09T17:36:02","slug":"vectorii-marimi-vectoriale","status":"publish","type":"post","link":"https:\/\/cedra.academy\/?p=2323","title":{"rendered":"Vectorii. M\u0103rimi vectoriale"},"content":{"rendered":"\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-06861a82-aff5-4979-8bdf-57ec343b446a\" href=\"https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marimile-vectoriale.pptx\">Marimile  vectoriale        PPTX                           .<\/a><a href=\"https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marimile-vectoriale.pptx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-06861a82-aff5-4979-8bdf-57ec343b446a\">Descarc\u0103<\/a><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\u00cen via\u021ba de zi cu zi \u0219i \u00een fizic\u0103 \u00eent\u00e2lnim m\u0103rimi care nu pot fi descrise complet doar printr-un num\u0103r. Acestea se numesc <strong>m\u0103rimi vectoriale<\/strong>. Spre deosebire de m\u0103rimile scalare, care au doar <strong>valoare numeric\u0103 \u0219i unitate de m\u0103sur\u0103<\/strong> (cum ar fi masa sau temperatura), m\u0103rimile vectoriale au <strong>valoare, direc\u021bie \u0219i sens<\/strong>. Exemple de m\u0103rimi vectoriale sunt <strong>viteza<\/strong>, <strong>for\u021ba<\/strong>, <strong>deplasarea<\/strong>, <strong>accelera\u021bia<\/strong> sau <strong>momentul de for\u021b\u0103<\/strong>.<\/p>\n\n\n\n<p>Un <strong>vector<\/strong> este o <strong>reprezentare matematic\u0103 a unei m\u0103rimi vectoriale<\/strong>. El este adesea ilustrat printr-o <strong>s\u0103geat\u0103<\/strong>:<\/p>\n\n\n\n<ul>\n<li><strong>lungimea s\u0103ge\u021bii<\/strong> indic\u0103 <strong>valoarea m\u0103rimii<\/strong>;<\/li>\n\n\n\n<li><strong>orientarea s\u0103ge\u021bii<\/strong> arat\u0103 <strong>direc\u021bia m\u0103rimii<\/strong>;<\/li>\n\n\n\n<li><strong>capul s\u0103ge\u021bii<\/strong> indic\u0103 <strong>sensul<\/strong> \u00een care ac\u021bioneaz\u0103 m\u0103rimea.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Elementele unui vector<\/strong><\/h4>\n\n\n\n<p>Fiecare vector are trei caracteristici principale:<\/p>\n\n\n\n<ol>\n<li><strong>Modulul (m\u0103rimea vectorului)<\/strong> \u2013 exprim\u0103 c\u00e2t de mare este m\u0103rimea;<\/li>\n\n\n\n<li><strong>Direc\u021bia<\/strong> \u2013 linia pe care se afl\u0103 vectorul \u00een spa\u021biu;<\/li>\n\n\n\n<li><strong>Sensul<\/strong> \u2013 orientarea vectorului de-a lungul direc\u021biei.<\/li>\n<\/ol>\n\n\n\n<p>De exemplu, dac\u0103 tragem o cutie cu o <strong>for\u021b\u0103 de 50 N<\/strong> spre nord, vectorul for\u021bei are modulul 50 N, direc\u021bia spre nord \u0219i sensul \u00eenspre locul \u00een care tragem cutia.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Reprezentarea vectorilor<\/strong><\/h4>\n\n\n\n<p>Vectorii pot fi reprezenta\u021bi \u00een mai multe moduri:<\/p>\n\n\n\n<ul>\n<li><strong>Grafic<\/strong> \u2013 printr-o s\u0103geat\u0103 desenat\u0103 pe h\u00e2rtie;<\/li>\n\n\n\n<li><strong>Analitic<\/strong> \u2013 folosind coordonate \u00een sistemul de axe (x, y, z), de exemplu: A\u20d7=(3,4)\\vec{A} = (3, 4)A=(3,4);<\/li>\n\n\n\n<li><strong>Notational<\/strong> \u2013 prin litere cu s\u0103geat\u0103 deasupra: v\u20d7\\vec{v}v, F\u20d7\\vec{F}F.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Opera\u021bii cu vectori<\/strong><\/h4>\n\n\n\n<p>Pe l\u00e2ng\u0103 reprezentarea lor, vectorii pot fi <strong>aduna\u021bi<\/strong>, <strong>sc\u0103zu\u021bi<\/strong> sau <strong>\u00eenmul\u021bi\u021bi<\/strong> cu un scalar:<\/p>\n\n\n\n<ul>\n<li><strong>Adunarea vectorilor<\/strong>: se face prin metoda paralelogramului sau metoda v\u00e2rfului la coad\u0103.<\/li>\n\n\n\n<li><strong>Sc\u0103derea vectorilor<\/strong>: se face prin adunarea vectorului opus.<\/li>\n\n\n\n<li><strong>\u00cenmul\u021birea cu un scalar<\/strong>: modific\u0103 doar modulul vectorului, p\u0103str\u00e2nd direc\u021bia, sau \u00eel inverseaz\u0103 dac\u0103 scalarul este negativ.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Exemple de vectori \u00een via\u021ba real\u0103<\/strong><\/h4>\n\n\n\n<ul>\n<li><strong>Viteza unei ma\u0219ini<\/strong>: are direc\u021bie \u0219i sens, nu doar valoare numeric\u0103;<\/li>\n\n\n\n<li><strong>Deplasarea unui elev pe holul \u0219colii<\/strong>: indic\u0103 distan\u021ba \u0219i direc\u021bia parcurgerii;<\/li>\n\n\n\n<li><strong>For\u021ba v\u00e2ntului asupra unei cor\u0103bii<\/strong>: direc\u021bia \u0219i sensul v\u00e2ntului influen\u021beaz\u0103 traiectoria navei.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Importan\u021ba vectorilor<\/strong><\/h4>\n\n\n\n<p>Vectorii sunt fundamentali \u00een fizic\u0103 \u0219i inginerie, deoarece majoritatea fenomenelor reale implic\u0103 <strong>m\u0103rimi care nu sunt doar cantitative<\/strong>, ci au \u0219i <strong>direc\u021bie \u0219i sens<\/strong>. \u00cen\u021belegerea vectorilor permite descrierea corect\u0103 a <strong>mi\u0219c\u0103rii obiectelor<\/strong>, <strong>for\u021belor aplicate<\/strong> \u0219i <strong>interac\u021biunilor<\/strong> din natur\u0103.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" loading=\"lazy\" width=\"478\" height=\"360\" src=\"https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Exemple_vectori-1.webp\" alt=\"\" class=\"wp-image-2326\" srcset=\"https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Exemple_vectori-1.webp 478w, https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Exemple_vectori-1-300x226.webp 300w\" sizes=\"(max-width: 478px) 100vw, 478px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Exerci\u021bii: Vectorii. M\u0103rimi vectoriale<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Exerci\u021biul 1 \u2013 Identificarea vectorilor<\/strong><\/h4>\n\n\n\n<p>Indic\u0103 care dintre urm\u0103toarele m\u0103rimi sunt <strong>vectoriale<\/strong> \u0219i care <strong>scalare<\/strong> (m\u0103rimi scalare):<\/p>\n\n\n\n<ol>\n<li>Viteza unei ma\u0219ini<\/li>\n\n\n\n<li>Temperatura aerului<\/li>\n\n\n\n<li>For\u021ba aplicat\u0103 asupra unei u\u0219i<\/li>\n\n\n\n<li>Masa unui obiect<\/li>\n\n\n\n<li>Desplasarea unui biciclist<\/li>\n\n\n\n<li>Energia unui corp<\/li>\n<\/ol>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Exerci\u021biul 2 \u2013 Reprezentarea vectorial\u0103<\/strong><\/h4>\n\n\n\n<p>Deseneaz\u0103 un vector care:<\/p>\n\n\n\n<ol>\n<li>Are <strong>lungimea 5 cm<\/strong> \u0219i direc\u021bia spre <strong>dreapta<\/strong><\/li>\n\n\n\n<li>Are <strong>lungimea 3 cm<\/strong>, orientat <strong>\u00een sus \u0219i la st\u00e2nga<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Men\u021bioneaz\u0103 <strong>modulul<\/strong>, <strong>direc\u021bia<\/strong> \u0219i <strong>sensul<\/strong> fiec\u0103rui vector.<\/p>\n\n\n\n<p>Subtitlu: modific\u0103 tipul sau stilul blocului<\/p>\n\n\n\n<p>Mut\u0103 blocul Subtitlu din pozi\u021bia 33 mai sus la pozi\u021bia 32<\/p>\n\n\n\n<p>Mut\u0103 blocul Subtitlu din pozi\u021bia 33 mai jos la pozi\u021bia 34<\/p>\n\n\n\n<p>Modific\u0103 alinierea<\/p>\n\n\n\n<p>Modific\u0103 alinierea textului<\/p>\n\n\n\n<p>Afi\u0219eaz\u0103 mai multe unelte pentru blocuri<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Exerci\u021biul 3 \u2013 Opera\u021bii cu vectori<\/strong><\/h4>\n\n\n\n<p>Se dau doi vectori \u00een planul cartezian:<br>A\u20d7=(3,4)\\vec{A} = (3, 4)A=(3,4), B\u20d7=(\u22121,2)\\vec{B} = (-1, 2)B=(\u22121,2)<\/p>\n\n\n\n<ol>\n<li>Calculeaz\u0103 suma vectorilor: A\u20d7+B\u20d7\\vec{A} + \\vec{B}A+B<\/li>\n\n\n\n<li>Calculeaz\u0103 diferen\u021ba vectorilor: A\u20d7\u2212B\u20d7\\vec{A} &#8211; \\vec{B}A\u2212B<\/li>\n\n\n\n<li>Determin\u0103 modulul vectorului A\u20d7\\vec{A}A \u0219i B\u20d7\\vec{B}B<\/li>\n<\/ol>\n\n\n\n<p><em>Not\u0103:<\/em> Modulul unui vector V\u20d7=(x,y)\\vec{V} = (x, y)V=(x,y) se calculeaz\u0103 cu formula: \u2223V\u20d7\u2223=x2+y2|\\vec{V}| = \\sqrt{x^2 + y^2}\u2223V\u2223=x2+y2\u200b<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Exerci\u021biul 4 \u2013 Vectori coliniari<\/strong><\/h4>\n\n\n\n<p>Se dau trei vectori:<br>U\u20d7=(2,4)\\vec{U} = (2, 4)U=(2,4), V\u20d7=(1,2)\\vec{V} = (1, 2)V=(1,2), W\u20d7=(\u22121,\u22122)\\vec{W} = (-1, -2)W=(\u22121,\u22122)<\/p>\n\n\n\n<ol>\n<li>Verific\u0103 dac\u0103 ace\u0219ti vectori sunt coliniari.<\/li>\n\n\n\n<li>Explic\u0103 ra\u021bionamentul folosit.<\/li>\n<\/ol>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Exerci\u021biul 5 \u2013 Aplicare \u00een probleme reale<\/strong><\/h4>\n\n\n\n<p>Un copil \u00eempinge un c\u0103rucior cu o <strong>for\u021b\u0103 de 50 N<\/strong> spre <strong>nord<\/strong>, iar un alt copil \u00eel \u00eempinge cu <strong>30 N<\/strong> spre <strong>est<\/strong>.<\/p>\n\n\n\n<ol>\n<li>Reprezint\u0103 grafic cei doi vectori.<\/li>\n\n\n\n<li>Determin\u0103 vectorul rezultant al for\u021belor (m\u0103rime \u0219i direc\u021bie).<\/li>\n<\/ol>\n\n\n\n<p><em>Hint:<\/em> Po\u021bi folosi teorema lui Pitagora \u0219i trigonometria pentru a afla direc\u021bia: Frez=F12+F22,\u03b1=arctan\u2061F2F1F_\\text{rez} = \\sqrt{F_1^2 + F_2^2}, \\quad \\alpha = \\arctan\\frac{F_2}{F_1}Frez\u200b=F12\u200b+F22\u200b\u200b,\u03b1=arctanF1\u200bF2\u200b\u200b<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" width=\"1024\" height=\"1024\" src=\"https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marime.png\" alt=\"\" class=\"wp-image-2328\" style=\"aspect-ratio:1;width:322px;height:auto\" srcset=\"https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marime.png 1024w, https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marime-300x300.png 300w, https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marime-150x150.png 150w, https:\/\/cedra.academy\/wp-content\/uploads\/2025\/09\/Marime-768x768.png 768w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>\u00cen via\u021ba de zi cu zi \u0219i \u00een fizic\u0103 \u00eent\u00e2lnim m\u0103rimi care nu pot fi descrise complet doar printr-un num\u0103r. Acestea se numesc m\u0103rimi vectoriale. Spre deosebire de m\u0103rimile scalare, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2332,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13,10],"tags":[],"_links":{"self":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts\/2323"}],"collection":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2323"}],"version-history":[{"count":2,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts\/2323\/revisions"}],"predecessor-version":[{"id":2331,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/posts\/2323\/revisions\/2331"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=\/wp\/v2\/media\/2332"}],"wp:attachment":[{"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2323"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2323"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cedra.academy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}