{"id":4086,"date":"2022-04-14T18:15:52","date_gmt":"2022-04-14T18:15:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-the-dot-product\/"},"modified":"2022-11-09T16:31:47","modified_gmt":"2022-11-09T16:31:47","slug":"skills-review-for-the-dot-product","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-the-dot-product\/","title":{"raw":"Skills Review for the Dot Product, Cross Product, and Equations of Lines and Planes in Space","rendered":"Skills Review for the Dot Product, Cross Product, and Equations of Lines and Planes in Space"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify reference angles for angles measured in both radians and degrees<\/li>\r\n \t<li>Evaluate the determinant of a [latex]2\\times2[\/latex] matrix<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Dot Product, Cross Product, and Equations of Lines and Planes in Space sections, we will explore the applications of and how to find the dot product and cross product of two vectors. Here we will review how to evaluate the sine and cosine functions at specific angle measures and evaluate the determinant of a [latex]2\\times2[\/latex] matrix.\r\n<h2>Find Reference Angles<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-vectors-in-the-plane\/\" target=\"_blank\" rel=\"noopener\">Module 2, Skills Review for Vectors in the Plane<\/a>)<\/em><\/strong>\r\n<h2>Evaluate the Determinant of a [latex]2\\times2[\/latex] Matrix<\/h2>\r\nA determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.\r\n<div class=\"textbox shaded\">\r\n<h3>A General Note: Find the Determinant of a [latex]2\\times2[\/latex] Matrix<\/h3>\r\nThe <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex]<\/div>\r\nis defined as\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\nNotice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\r\nFind the determinant of the given matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"149250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149250\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&amp;=\\left\\rvert\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right\\rvert\\hfill \\\\ &amp;=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &amp;=27\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]6397[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify reference angles for angles measured in both radians and degrees<\/li>\n<li>Evaluate the determinant of a [latex]2\\times2[\/latex] matrix<\/li>\n<\/ul>\n<\/div>\n<p>In the Dot Product, Cross Product, and Equations of Lines and Planes in Space sections, we will explore the applications of and how to find the dot product and cross product of two vectors. Here we will review how to evaluate the sine and cosine functions at specific angle measures and evaluate the determinant of a [latex]2\\times2[\/latex] matrix.<\/p>\n<h2>Find Reference Angles<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-vectors-in-the-plane\/\" target=\"_blank\" rel=\"noopener\">Module 2, Skills Review for Vectors in the Plane<\/a>)<\/em><\/strong><\/p>\n<h2>Evaluate the Determinant of a [latex]2\\times2[\/latex] Matrix<\/h2>\n<p>A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.<\/p>\n<div class=\"textbox shaded\">\n<h3>A General Note: Find the Determinant of a [latex]2\\times2[\/latex] Matrix<\/h3>\n<p>The <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex]<\/div>\n<p>is defined as<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p>Notice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p>Find the determinant of the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149250\">Show Solution<\/span><\/p>\n<div id=\"q149250\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&=\\left\\rvert\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right\\rvert\\hfill \\\\ &=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &=27\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6397\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6397&theme=oea&iframe_resize_id=ohm6397\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4086\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus1\/\">https:\/\/courses.lumenlearning.com\/calculus1\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Calculus Volume 2. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus2\/\">https:\/\/courses.lumenlearning.com\/calculus2\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/calculus1\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/calculus2\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4086","chapter","type-chapter","status-publish","hentry"],"part":4109,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4086","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4086\/revisions"}],"predecessor-version":[{"id":4715,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4086\/revisions\/4715"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/4109"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4086\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=4086"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=4086"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=4086"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=4086"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}