{"id":504,"date":"2025-10-09T15:15:59","date_gmt":"2025-10-09T15:15:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=504"},"modified":"2025-11-17T14:32:00","modified_gmt":"2025-11-17T14:32:00","slug":"vector-operations-in-space-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-operations-in-space-apply-it\/","title":{"raw":"Vector Operations in Space: Apply It","rendered":"Vector Operations in Space: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate the dot product of two given vectors<\/li>\r\n \t<li>Find angles between vectors and determine when vectors are perpendicular using dot products<\/li>\r\n \t<li>Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it<\/li>\r\n \t<li>Calculate the work done by a given force<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Abstract Vector Applications<\/h3>\r\n<p class=\"whitespace-normal break-words\">Vectors aren't limited to physical quantities like force and position. They're powerful tools for organizing and processing data.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Consider a fruit vendor who sells apples, bananas, and oranges. He can represent his daily sales with a <strong>quantity vector<\/strong> [latex]\\mathbf{q} = \\langle 30, 12, 18 \\rangle[\/latex] ([latex]30[\/latex] apples, [latex]12[\/latex] bananas, [latex]18[\/latex] oranges) and his prices with a <strong>price vector<\/strong> [latex]\\mathbf{p} = \\langle 0.50, 0.25, 1.00 \\rangle[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">The dot product [latex]\\mathbf{q} \\cdot \\mathbf{p} = 30(0.50) + 12(0.25) + 18(1.00) = 15 + 3 + 18 = 36[\/latex] gives his total daily revenue: [latex]$36[\/latex].<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">This approach extends to any number of dimensions. If the vendor adds grapefruit, he simply uses four-dimensional vectors. The dot product calculation follows the same pattern: multiply corresponding components and add the results.<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Vectors provide an elegant way to organize related data. The dot product can compute totals, weighted averages, and correlations between datasets.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1163724036566\">AAA Party Supply Store sells invitations, party favors, decorations, and food service items such as paper plates and napkins. When AAA buys its inventory, it pays [latex]25\u00a2[\/latex] per package for invitations and party favors. Decorations cost AAA [latex]50\u00a2[\/latex] each, and food service items cost [latex]20\u00a2[\/latex] per package. AAA sells invitations for [latex]$2.50[\/latex] per package and party favors for [latex]$1.50[\/latex] per package. Decorations sell for [latex]$4.50[\/latex] each and food service items for [latex]$1.25[\/latex] per package.<\/p>\r\n<p id=\"fs-id1163724036578\">During the month of May, AAA Party Supply Store sells [latex]1258[\/latex] invitations, [latex]342[\/latex] party favors, [latex]2426[\/latex] decorations, and [latex]1354[\/latex] food service items. Use vectors and dot products to calculate how much money AAA made in sales during the month of May. How much did the store make in profit?<\/p>\r\n[reveal-answer q=\"734614322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"734614322\"]\r\n<p id=\"fs-id1163724036587\">The cost, price, and quantity vectors are<\/p>\r\n\r\n<center>\r\n[latex]\r\n\\begin{align*}\r\n\\mathbf{c} &amp;= \\langle 0.25,0.25,0.50,0.20 \\rangle\\\\\r\n\\mathbf{p}&amp;= \\langle 2.50,1.50,4.50,1.25 \\rangle\\\\\r\n\\mathbf{q}&amp;= \\langle 1258,342,2426,1354 \\rangle.\\\\\\end{align*}\r\n[\/latex]<\/center>\r\n<p id=\"fs-id1163724119482\">AAA sales for the month of May can be calculated using the dot product [latex]\\mathbf{p} \\cdot \\mathbf{q}[\/latex]. We have<\/p>\r\n\r\n<center>\r\n[latex]\r\n\\begin{align*}\r\n\\mathbf{p} \\cdot \\mathbf{q} &amp;= \\langle 2.50,1.50,4.50,1.25 \\rangle \\cdot \\langle 1258,342,2426,1354 \\rangle\\\\\r\n&amp;= 3145+513+10917+1692.5\\\\\r\n&amp;= 16267.5.\\\\\\end{align*}\r\n[\/latex]<\/center>\r\n<p id=\"fs-id1163723283999\">So, AAA took in [latex]$16,267.50[\/latex] during the month of May.<\/p>\r\n<p id=\"fs-id1163723284003\">To calculate the profit, we must first calculate how much AAA paid for the items sold. We use the dot product [latex]\\mathbf{c} \\cdot \\mathbf{q}[\/latex] to get<\/p>\r\n\r\n<center>\r\n[latex]\r\n\\begin{align*}\r\n\\mathbf{c} \\cdot \\mathbf{q} &amp;= \\langle 0.25,0.25,0.50,0.20 \\rangle \\cdot \\langle 1258,342,2426,1354 \\rangle\\\\\r\n&amp;= 314.5+85.5+1213+270.8\\\\\r\n&amp;= 1883.8.\\\\\\end{align*}\r\n[\/latex]<\/center>\r\n<p id=\"fs-id1163723326660\">So, AAA paid [latex]$1,883.80 [\/latex]for the items they sold. Their profit, then, is given by<\/p>\r\n\r\n<center>\r\n[latex]\r\n\\begin{align*}\r\n\\mathbf{p} \\cdot \\mathbf{q} - \\mathbf{c} \\cdot \\mathbf{q} &amp;= 16267.5-1883.8\\\\\r\n&amp;= 14383.7.\\\\\\end{align*}\r\n[\/latex]<\/center>\r\n<p id=\"fs-id1163723326731\">Therefore, AAA Party Supply Store made [latex]$14,383.70[\/latex] in May.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">On June 1, AAA Party Supply Store decided to increase the price they charge for party favors to [latex]$2 [\/latex]per package. They also changed suppliers for their invitations, and are now able to purchase invitations for only [latex]10\u00a2[\/latex] per package. All their other costs and prices remain the same.\r\n[latex]\\\\[\/latex]\r\nIf AAA sells [latex]1408[\/latex] invitations, [latex]147[\/latex] party favors, [latex]2112[\/latex] decorations, and [latex]1894[\/latex] food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.[reveal-answer q=\"000034625\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"000034625\"]Sales = [latex]$15,685.50[\/latex]; profit = [latex]$14,073.15[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7753521&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=FzU_D0eXcYs&amp;video_target=tpm-plugin-9rztup8m-FzU_D0eXcYs\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.26_transcript.html\">\u201cCP 2.26\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate the dot product of two given vectors<\/li>\n<li>Find angles between vectors and determine when vectors are perpendicular using dot products<\/li>\n<li>Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it<\/li>\n<li>Calculate the work done by a given force<\/li>\n<\/ul>\n<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Abstract Vector Applications<\/h3>\n<p class=\"whitespace-normal break-words\">Vectors aren&#8217;t limited to physical quantities like force and position. They&#8217;re powerful tools for organizing and processing data.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Consider a fruit vendor who sells apples, bananas, and oranges. He can represent his daily sales with a <strong>quantity vector<\/strong> [latex]\\mathbf{q} = \\langle 30, 12, 18 \\rangle[\/latex] ([latex]30[\/latex] apples, [latex]12[\/latex] bananas, [latex]18[\/latex] oranges) and his prices with a <strong>price vector<\/strong> [latex]\\mathbf{p} = \\langle 0.50, 0.25, 1.00 \\rangle[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">The dot product [latex]\\mathbf{q} \\cdot \\mathbf{p} = 30(0.50) + 12(0.25) + 18(1.00) = 15 + 3 + 18 = 36[\/latex] gives his total daily revenue: [latex]$36[\/latex].<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">This approach extends to any number of dimensions. If the vendor adds grapefruit, he simply uses four-dimensional vectors. The dot product calculation follows the same pattern: multiply corresponding components and add the results.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Vectors provide an elegant way to organize related data. The dot product can compute totals, weighted averages, and correlations between datasets.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1163724036566\">AAA Party Supply Store sells invitations, party favors, decorations, and food service items such as paper plates and napkins. When AAA buys its inventory, it pays [latex]25\u00a2[\/latex] per package for invitations and party favors. Decorations cost AAA [latex]50\u00a2[\/latex] each, and food service items cost [latex]20\u00a2[\/latex] per package. AAA sells invitations for [latex]$2.50[\/latex] per package and party favors for [latex]$1.50[\/latex] per package. Decorations sell for [latex]$4.50[\/latex] each and food service items for [latex]$1.25[\/latex] per package.<\/p>\n<p id=\"fs-id1163724036578\">During the month of May, AAA Party Supply Store sells [latex]1258[\/latex] invitations, [latex]342[\/latex] party favors, [latex]2426[\/latex] decorations, and [latex]1354[\/latex] food service items. Use vectors and dot products to calculate how much money AAA made in sales during the month of May. How much did the store make in profit?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q734614322\">Show Solution<\/button><\/p>\n<div id=\"q734614322\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163724036587\">The cost, price, and quantity vectors are<\/p>\n<div style=\"text-align: center;\">\n[latex]\\begin{align*}  \\mathbf{c} &= \\langle 0.25,0.25,0.50,0.20 \\rangle\\\\  \\mathbf{p}&= \\langle 2.50,1.50,4.50,1.25 \\rangle\\\\  \\mathbf{q}&= \\langle 1258,342,2426,1354 \\rangle.\\\\\\end{align*}[\/latex]<\/div>\n<p id=\"fs-id1163724119482\">AAA sales for the month of May can be calculated using the dot product [latex]\\mathbf{p} \\cdot \\mathbf{q}[\/latex]. We have<\/p>\n<div style=\"text-align: center;\">\n[latex]\\begin{align*}  \\mathbf{p} \\cdot \\mathbf{q} &= \\langle 2.50,1.50,4.50,1.25 \\rangle \\cdot \\langle 1258,342,2426,1354 \\rangle\\\\  &= 3145+513+10917+1692.5\\\\  &= 16267.5.\\\\\\end{align*}[\/latex]<\/div>\n<p id=\"fs-id1163723283999\">So, AAA took in [latex]$16,267.50[\/latex] during the month of May.<\/p>\n<p id=\"fs-id1163723284003\">To calculate the profit, we must first calculate how much AAA paid for the items sold. We use the dot product [latex]\\mathbf{c} \\cdot \\mathbf{q}[\/latex] to get<\/p>\n<div style=\"text-align: center;\">\n[latex]\\begin{align*}  \\mathbf{c} \\cdot \\mathbf{q} &= \\langle 0.25,0.25,0.50,0.20 \\rangle \\cdot \\langle 1258,342,2426,1354 \\rangle\\\\  &= 314.5+85.5+1213+270.8\\\\  &= 1883.8.\\\\\\end{align*}[\/latex]<\/div>\n<p id=\"fs-id1163723326660\">So, AAA paid [latex]$1,883.80[\/latex]for the items they sold. Their profit, then, is given by<\/p>\n<div style=\"text-align: center;\">\n[latex]\\begin{align*}  \\mathbf{p} \\cdot \\mathbf{q} - \\mathbf{c} \\cdot \\mathbf{q} &= 16267.5-1883.8\\\\  &= 14383.7.\\\\\\end{align*}[\/latex]<\/div>\n<p id=\"fs-id1163723326731\">Therefore, AAA Party Supply Store made [latex]$14,383.70[\/latex] in May.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">On June 1, AAA Party Supply Store decided to increase the price they charge for party favors to [latex]$2[\/latex]per package. They also changed suppliers for their invitations, and are now able to purchase invitations for only [latex]10\u00a2[\/latex] per package. All their other costs and prices remain the same.<br \/>\n[latex]\\\\[\/latex]<br \/>\nIf AAA sells [latex]1408[\/latex] invitations, [latex]147[\/latex] party favors, [latex]2112[\/latex] decorations, and [latex]1894[\/latex] food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q000034625\">Show Solution<\/button><\/p>\n<div id=\"q000034625\" class=\"hidden-answer\" style=\"display: none\">Sales = [latex]$15,685.50[\/latex]; profit = [latex]$14,073.15[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7753521&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=FzU_D0eXcYs&amp;video_target=tpm-plugin-9rztup8m-FzU_D0eXcYs\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.26_transcript.html\">\u201cCP 2.26\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":20,"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-504","chapter","type-chapter","status-publish","hentry"],"part":420,"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\/504","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":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/504\/revisions"}],"predecessor-version":[{"id":2281,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/504\/revisions\/2281"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/420"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/504\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=504"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=504"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=504"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=504"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}