{"id":521,"date":"2025-10-09T15:17:33","date_gmt":"2025-10-09T15:17:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=521"},"modified":"2025-12-04T19:36:56","modified_gmt":"2025-12-04T19:36:56","slug":"cross-product-and-applications-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/cross-product-and-applications-apply-it\/","title":{"raw":"Cross Product and Applications: Apply It","rendered":"Cross Product and Applications: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate the cross product of two vectors and understand what it tells us<\/li>\r\n \t<li>Find a vector orthogonal to two given vectors<\/li>\r\n \t<li>Determine areas and volumes by using the cross product<\/li>\r\n \t<li>Calculate the torque of a given force and position vector<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Building a Community Garden with Cross Products<\/h2>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">You're volunteering with a community group to design a small neighborhood garden on a triangular plot of land. The garden committee has measured the plot and provided you with coordinate data for the three corner posts. Your tasks include calculating the area of the garden space to order the right amount of soil and mulch, determining which direction to install a perpendicular drainage pipe, and figuring out the torque needed to install support posts using hand tools.<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The three corner posts of the triangular plot are located at points [latex]P = (2, 1, 0)[\/latex], [latex]Q = (5, 3, 0)[\/latex], and [latex]R = (3, 5, 0)[\/latex], where all measurements are in meters. You'll use vectors and cross products to analyze this space and make practical decisions for the garden project.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">All of your work will help the committee purchase materials efficiently and plan the installation process.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\r\n\r\n[ohm_question hide_question_numbers=1]316524[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316547[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316548[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]316551[\/ohm_question]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate the cross product of two vectors and understand what it tells us<\/li>\n<li>Find a vector orthogonal to two given vectors<\/li>\n<li>Determine areas and volumes by using the cross product<\/li>\n<li>Calculate the torque of a given force and position vector<\/li>\n<\/ul>\n<\/section>\n<h2>Building a Community Garden with Cross Products<\/h2>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">You&#8217;re volunteering with a community group to design a small neighborhood garden on a triangular plot of land. The garden committee has measured the plot and provided you with coordinate data for the three corner posts. Your tasks include calculating the area of the garden space to order the right amount of soil and mulch, determining which direction to install a perpendicular drainage pipe, and figuring out the torque needed to install support posts using hand tools.<\/div>\n<\/div>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">The three corner posts of the triangular plot are located at points [latex]P = (2, 1, 0)[\/latex], [latex]Q = (5, 3, 0)[\/latex], and [latex]R = (3, 5, 0)[\/latex], where all measurements are in meters. You&#8217;ll use vectors and cross products to analyze this space and make practical decisions for the garden project.<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">All of your work will help the committee purchase materials efficiently and plan the installation process.<\/p>\n<\/div>\n<\/div>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0\">\n<p><iframe loading=\"lazy\" id=\"ohm316524\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316524&theme=lumen&iframe_resize_id=ohm316524&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316547\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316547&theme=lumen&iframe_resize_id=ohm316547&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316548\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316548&theme=lumen&iframe_resize_id=ohm316548&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm316551\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316551&theme=lumen&iframe_resize_id=ohm316551&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-521","chapter","type-chapter","status-publish","hentry"],"part":422,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/521","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/521\/revisions"}],"predecessor-version":[{"id":2753,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/521\/revisions\/2753"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/422"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/521\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=521"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=521"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=521"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=521"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}