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The operations of addition, subtraction and multiplication familiar in the algebra of numbers are, with suitable definition, capable of extension to an algebra of vectors. The following definitions are fundamental. <\/p>\n
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Two vectors $\\bold{A}$ and $\\bold{B}$ are equal<\/em> if they have the same magnitude and direction regardless of their initial points. <\/li>\n
A vector having direction opposite to that of vector $\\bold{A}$ but with the same magnitude is denoted by $-\\bold{A}$ . <\/li>\n
The sum<\/em> or resultant<\/em> of vectors $\\bold{A}$ and $\\bold{B}$ is a vector $\\bold{C}$ formed by placing the initial point of $\\bold{B}$ on the terminal point of $\\bold{A}$ and joining the initial point of $\\bold{A}$ to the terminal point of $\\bold{B}$ . The sum $\\bold{C}$ is written $\\bold{C} = \\bold{A} + \\bold{B}$ . The definition here is equivalent to the parallelogram law<\/em> for vector addition. <\/li>\n
The difference<\/em> of vectors $\\bold{A}$ and $\\bold{B}$ , represented by $\\bold{A} - \\bold{B}$ , is that vector $\\bold{C}$ which added to $\\bold{B}$ gives $\\bold{A}$ . Equivalently, $\\bold{A} - \\bold{B}$ may be defined as $\\bold{A} + (-\\bold{B})$ . If $\\bold{A} = \\bold{B}$ , then $\\bold{A} - \\bold{B}$ is defined as the null<\/em> or zero vector<\/em> and is represented by the symbol $\\bold{0}$ . This has a magnitude of zero but its direction is not defined. <\/li>\n
Multiplication of vector $\\bold{A}$ by a scalar m<\/em> produces a vector $m\\bold{A}$ with magnitude $|m|$ times the magnitude of $\\bold{A}$ and direction the same as or opposite to that of $\\bold{A}$ according as $m$ is positive or negative. If $m = 0$ , $m\\bold{A} = \\bold{0}$ , the null vector. <\/li>\n<\/ol>\n<\/blockquote>\n
<\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>The operations of addition, subtraction and multiplication familiar in the algebra of numbers are, with suitab … <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4870\" class=\"more-link\"><span class=\"screen-reader-text\">“Vector algebra” \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-4870","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4870","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4870"}],"version-history":[{"count":9,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4870\/revisions"}],"predecessor-version":[{"id":6045,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4870\/revisions\/6045"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4870"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4870"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4870"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}