{"id":207,"date":"2017-08-23T10:00:01","date_gmt":"2017-08-23T10:00:01","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=207"},"modified":"2017-09-18T19:43:13","modified_gmt":"2017-09-18T19:43:13","slug":"special-matrix-products","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/special-matrix-products\/","title":{"rendered":"Special Matrix Products"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometric-interpretations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Next Workshop \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Special Matrix Products<\/h1>\n<h4>Objective<\/h4>\n<ul>\n<li>A <em><strong>vector\u00a0<\/strong><\/em>is a matrix that has either a single column or a single row.<\/li>\n<li>The length of a vector,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image003.gif\" width=\"168\" height=\"26\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0is denoted\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p1_clip_image006.gif\" width=\"19\" height=\"24\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and calculated<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p1_clip_image009.gif\" width=\"177\" height=\"30\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <strong><em>Law of Cosines<\/em><\/strong> indicates\u00a0that in any triangle <em>ABC<\/em>,<\/li>\n<\/ul>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p1_clip_image012.gif\" width=\"157\" height=\"73\" name=\"graphics5\" align=\"MIDDLE\" border=\"0\" \/>.<\/p>\n<section>\n<h3>What is a dot product?<\/h3>\n<p>A <abbr title=\" the sum produced when corresponding elements of two vectors are multiplied together and added. The two vectors must have the same dimensions. \">dot product<\/abbr> of two vectors,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image003.gif\" width=\"168\" height=\"26\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image006.gif\" width=\"164\" height=\"26\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0is denoted\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image009.gif\" width=\"30\" height=\"14\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and is defined: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image012.gif\" width=\"221\" height=\"24\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/><\/p>\n<p>In two dimensions, a vector can be graphed onto an\u00a0<em><span style=\"text-decoration: none;\">xy<\/span><\/em><span style=\"text-decoration: none;\">-coor<\/span>dinate\u00a0plane.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.4%20Art%20007.JPG\" alt=\"Three Vectors\" width=\"144\" height=\"116\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The graph shows two vectors <strong><span style=\"color: #e0c907;\"><em><span style=\"text-decoration: none;\">u<\/span><\/em><span style=\"text-decoration: none;\"> = [2 3]<\/span><\/span><\/strong><span style=\"text-decoration: none;\"> and\u00a0<\/span><strong><span style=\"color: #ff2b05;\"><em><span style=\"text-decoration: none;\">v\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= [4 1]<\/span><\/span><\/strong><span style=\"text-decoration: none;\">. <\/span>The\u00a0blue vector is described as <strong><span style=\"color: #005df9;\"><em><span style=\"text-decoration: none;\">u\u00a0<\/span><\/em><span style=\"text-decoration: none;\">\u2013 <\/span><em><span style=\"text-decoration: none;\">v<\/span><\/em><\/span><\/strong><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p align=\"LEFT\">By the Law of Cosines,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image015.gif\" width=\"211\" height=\"28\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image018.gif\" width=\"16\" height=\"19\" name=\"graphics9\" align=\"MIDDLE\" border=\"0\" \/>\u00a0is the interior angle between <em>u<\/em> and <em>v<\/em>.<\/p>\n<p>Simplifying:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image021.gif\" width=\"316\" height=\"181\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Thus, the dot product is the product of the lengths of the two\u00a0vectors and the cosine of the angle between the vectors. More\u00a0helpfully, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image024.gif\" width=\"89\" height=\"46\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0provided <em>u<\/em> and <em>v <\/em>are non-zero vectors.<\/p>\n<p>For example, to find the angle between the two vectors <em>u\u00a0<\/em>= [2 3] and <em>v<\/em> = [4 1],<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image027.gif\" width=\"290\" height=\"56\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p2_clip_image030.gif\" width=\"102\" height=\"44\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>If the dot product of two non-zero vectors is 0, then what must\u00a0be true about the vectors?<\/p>\n<ol>\n<li>The vectors are perpendicular.<\/li>\n<li>The vectors are equal to each other.<\/li>\n<li>One vector is a scalar multiple of the other.<\/li>\n<li>The vectors do not exist.<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A. If\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s8_p2_html_m3e619d19.gif\" width=\"36\" height=\"15\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>=\u00a00, and the vectors are non-zero, then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s8_p2_html_m2bd113bc.gif\" width=\"92\" height=\"47\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0When the cosine is 0, the angle must be 90\u00b0.<\/p>\n<\/div>\n<\/section>\n<h4>What are Cross Products?<\/h4>\n<p>We define the cross product only for three-dimensional vectors.<\/p>\n<p><span style=\"text-decoration: none;\">Let <\/span><em><span style=\"text-decoration: none;\">u\u00a0<\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image003.gif\" width=\"87\" height=\"25\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and <\/span><em><span style=\"text-decoration: none;\">v\u00a0<\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image006.gif\" width=\"84\" height=\"25\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<p>And let <em>i<\/em>, <em>j<\/em>, and <em>k<\/em> be the unit vectors [1\u00a00 0], [0 1 0], and [0 0 1] respectively.<\/p>\n<p><span style=\"text-decoration: none;\">The <\/span><abbr title=\"The cross-product u \u00d7 v is a vector perpendicular to the plane in which both u and v lie. \"><span style=\"text-decoration: none;\">cross\u00a0product<\/span><\/abbr><strong><span style=\"text-decoration: none;\"> <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image009.gif\" width=\"52\" height=\"15\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">is the determinant\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image012.gif\" width=\"81\" height=\"73\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>The cross product is a vector. It is perpendicular\u00a0to the plane on which the original two vectors lie. So the cross\u00a0product <em>u<\/em> \u00d7 <em>v <\/em>is a vector perpendicular to the\u00a0plane in which both <em>u<\/em> and <em>v<\/em> lie.<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the cross product of <em>u<\/em> = [4 \u20132 3] and <em>v<\/em> = [\u20131 2 1]?<\/p>\n<ol>\n<li>\u20135<\/li>\n<li>\u201348<\/li>\n<li>3<em>i<\/em> + 0<em>j<\/em> + 4<em>k<\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image018.gif\" width=\"91\" height=\"21\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. The cross product yields the vector\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image021.gif\" width=\"389\" height=\"73\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h4>Does the cross product have a special geometric interpretation?<\/h4>\n<ul>\n<li>The length of the cross product\u00a0of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image024.gif\" width=\"52\" height=\"15\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the area of a parallelogram that has sides of <em>u<\/em> and <em>v.<\/em><\/li>\n<li>The length of the cross product of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s8_p3_clip_image027.gif\" width=\"88\" height=\"15\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the volume of the parallelepiped created by <em>u<\/em>, <em>v<\/em>,\u00a0and <em>w<\/em>.<\/li>\n<\/ul>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>The number of solutions of a\u00a0system of linear equations can be determined by graphing the\u00a0system.<\/li>\n<li>The number of solutions of a\u00a0system of linear equations is also evident by the symbolic\u00a0manipulations of the equations.<\/li>\n<li>A three-variable linear system\u00a0with one solution can be graphed as three planes meeting at\u00a0exactly one point.<\/li>\n<li>A three-variable linear system\u00a0with no solution can be graphed as three planes that never\u00a0coincide at the same time.<\/li>\n<li>A three-variable linear system\u00a0with an infinite number of solutions can be graphed as two or\u00a0three planes that meet at a line.<\/li>\n<li>A two-variable linear system\u00a0with one solution can be graphed as two lines that meet at\u00a0exactly one point.<\/li>\n<li>A two-variable linear system\u00a0with no solution can be graphed as two parallel lines that, of\u00a0course, never cross.<\/li>\n<li>A two-variable linear system\u00a0with an infinite number of solutions is graphed as one line\u00a0because both equations describe the same line.<\/li>\n<li>Determinants can be used in a\u00a0number of geometric applications.<\/li>\n<li>A <em><strong>dot product\u00a0<\/strong><\/em>between two vectors produces a <strong><em>scalar<\/em>.<\/strong><\/li>\n<li>A <strong><em>cross product\u00a0<\/em><\/strong>between two vectors produces a <strong><em>vector<\/em>. <\/strong><\/li>\n<li>Using the <strong><em>Law of Cosines<\/em>, <\/strong>we\u00a0developed a relation using the dot product and the length of the<br \/>\ntwo vectors to determine the angle between two vectors.<\/li>\n<\/ul>\n<h3>Further Reading in Linear Algebra<\/h3>\n<p><em>Linear Algebra and its Applications<\/em> (David C. Lay):\u00a0Pearson Addison Wesley, 2002.<\/p>\n<p><em>3,000 Solved Linear Algebra Problems<\/em> (Seymour\u00a0Lipschutz): McGraw Hill, 1989.<\/p>\n<p align=\"CENTER\"><strong><em>Don&#8217;t forget to test your knowledge\u00a0with the <a href=\"http:\/\/www.abcte.org\/drupal\/courses\/mrc\/quizzes\/lalgebra\" target=\"popsome\">Measurement and Linear Algebra Chapter Quiz; <\/a><\/em><\/strong><\/p>\n<\/section>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometric-interpretations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Next Workshop \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Workshop \u27a1 Special Matrix Products Objective A vector\u00a0is a matrix that has either a single column or a single row. The length of a vector,\u00a0,\u00a0is denoted\u00a0\u00a0and calculated. The Law of Cosines indicates\u00a0that in any triangle ABC, . What is a dot product? A dot product of two vectors,\u00a0and\u00a0,\u00a0is denoted\u00a0\u00a0and is defined: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-207","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=207"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/207\/revisions"}],"predecessor-version":[{"id":699,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/207\/revisions\/699"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}