{"id":4096,"date":"2022-04-14T18:15:53","date_gmt":"2022-04-14T18:15:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-directional-derivatives-and-the-gradient\/"},"modified":"2022-11-09T16:38:31","modified_gmt":"2022-11-09T16:38:31","slug":"skills-review-for-directional-derivatives-and-the-gradient","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-directional-derivatives-and-the-gradient\/","title":{"raw":"Skills Review for Directional Derivatives and the Gradient","rendered":"Skills Review for Directional Derivatives and the Gradient"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate trigonometric functions using the unit circle<\/li>\r\n \t<li>Find a unit vector in the direction of v<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Directional Derivatives and the Gradient section, we will learn how to further apply partial derivatives to directional derivatives and the gradient. Here we will review how to evaluate trigonometric functions and find a unit vector.\r\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-functions-of-several-variables\/\" target=\"_blank\" rel=\"noopener\">Module 4, Skills Review for Functions of Several Variables and Limits and Continuity<\/a>)<\/em><\/strong>\r\n<h2>Finding a Unit Vector<\/h2>\r\nWe call a vector with a magnitude of 1 a <strong>unit vector<\/strong>. This allows us to preserve the direction of the original vector while simplifying calculations.\r\n\r\nUnit vectors are defined in terms of components. The horizontal unit vector is written as [latex]\\boldsymbol{i}=\\langle 1,0\\rangle [\/latex] and is directed along the positive horizontal axis. The vertical unit vector is written as [latex]\\boldsymbol{j}=\\langle 0,1\\rangle [\/latex] and is directed along the positive vertical axis.\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\/27181159\/CNX_Precalc_Figure_08_08_011n2.jpg\" alt=\"Plot showing the unit vectors i=91,0) and j=(0,1)\" width=\"487\" height=\"253\" \/> <b>Figure 14<\/b>[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Unit Vectors<\/h3>\r\nIf [latex]\\boldsymbol{v}[\/latex] is a nonzero vector, then [latex]\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}[\/latex] is a unit vector in the direction of [latex]\\boldsymbol{v}[\/latex]. Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Unit Vector in the Direction of <em>v<\/em><\/h3>\r\nFind a unit vector in the same direction as [latex]\\boldsymbol{v}=\\langle -5,12\\rangle [\/latex].\r\n\r\n[reveal-answer q=\"326943\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"326943\"]\r\n\r\nFirst, we will find the magnitude.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}|\\boldsymbol{v}|&amp;=\\sqrt{{\\left(-5\\right)}^{2}+{\\left(12\\right)}^{2}} \\\\ &amp;=\\sqrt{25+144} \\\\ &amp;=\\sqrt{169} \\\\ &amp;=13\\end{align}[\/latex]<\/p>\r\nThen we divide each component by [latex]|v|[\/latex], which gives a unit vector in the same direction as <strong>v<\/strong>:\r\n<p style=\"text-align: center;\">[latex]\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}=-\\frac{5}{13}i+\\frac{12}{13}j[\/latex]<\/p>\r\nor, in component form\r\n<p style=\"text-align: center;\">[latex]\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}=\\langle -\\frac{5}{13},\\frac{12}{13}\\rangle [\/latex]<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181202\/CNX_Precalc_Figure_08_08_0122.jpg\" alt=\"Plot showing the unit vector (-5\/13, 12\/13) in the direction of (-5, 12)\" width=\"487\" height=\"628\" \/>\r\n\r\nVerify that the magnitude of the unit vector equals 1. The magnitude of [latex]-\\frac{5}{13}\\boldsymbol{i}+\\frac{12}{13}\\boldsymbol{j}[\/latex] is given as\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{{\\left(-\\frac{5}{13}\\right)}^{2}+{\\left(\\frac{12}{13}\\right)}^{2}}&amp;=\\sqrt{\\frac{25}{169}+\\frac{144}{169}} \\\\ &amp;=\\sqrt{\\frac{169}{169}}=1 \\end{align}[\/latex]<\/p>\r\nThe vector\u00a0[latex]u=\\frac{5}{13}[\/latex] <strong><em>i<\/em><\/strong> [latex]+\\frac{12}{13}[\/latex] <strong><em>j<\/em><\/strong> is the unit vector in the same direction as <strong><em>v<\/em><\/strong> [latex]=\\langle -5,12\\rangle [\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate trigonometric functions using the unit circle<\/li>\n<li>Find a unit vector in the direction of v<\/li>\n<\/ul>\n<\/div>\n<p>In the Directional Derivatives and the Gradient section, we will learn how to further apply partial derivatives to directional derivatives and the gradient. Here we will review how to evaluate trigonometric functions and find a unit vector.<\/p>\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-functions-of-several-variables\/\" target=\"_blank\" rel=\"noopener\">Module 4, Skills Review for Functions of Several Variables and Limits and Continuity<\/a>)<\/em><\/strong><\/p>\n<h2>Finding a Unit Vector<\/h2>\n<p>We call a vector with a magnitude of 1 a <strong>unit vector<\/strong>. This allows us to preserve the direction of the original vector while simplifying calculations.<\/p>\n<p>Unit vectors are defined in terms of components. The horizontal unit vector is written as [latex]\\boldsymbol{i}=\\langle 1,0\\rangle[\/latex] and is directed along the positive horizontal axis. The vertical unit vector is written as [latex]\\boldsymbol{j}=\\langle 0,1\\rangle[\/latex] and is directed along the positive vertical axis.<\/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\/27181159\/CNX_Precalc_Figure_08_08_011n2.jpg\" alt=\"Plot showing the unit vectors i=91,0) and j=(0,1)\" width=\"487\" height=\"253\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: The Unit Vectors<\/h3>\n<p>If [latex]\\boldsymbol{v}[\/latex] is a nonzero vector, then [latex]\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}[\/latex] is a unit vector in the direction of [latex]\\boldsymbol{v}[\/latex]. Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Unit Vector in the Direction of <em>v<\/em><\/h3>\n<p>Find a unit vector in the same direction as [latex]\\boldsymbol{v}=\\langle -5,12\\rangle[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q326943\">Show Solution<\/span><\/p>\n<div id=\"q326943\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we will find the magnitude.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}|\\boldsymbol{v}|&=\\sqrt{{\\left(-5\\right)}^{2}+{\\left(12\\right)}^{2}} \\\\ &=\\sqrt{25+144} \\\\ &=\\sqrt{169} \\\\ &=13\\end{align}[\/latex]<\/p>\n<p>Then we divide each component by [latex]|v|[\/latex], which gives a unit vector in the same direction as <strong>v<\/strong>:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}=-\\frac{5}{13}i+\\frac{12}{13}j[\/latex]<\/p>\n<p>or, in component form<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}=\\langle -\\frac{5}{13},\\frac{12}{13}\\rangle[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181202\/CNX_Precalc_Figure_08_08_0122.jpg\" alt=\"Plot showing the unit vector (-5\/13, 12\/13) in the direction of (-5, 12)\" width=\"487\" height=\"628\" \/><\/p>\n<p>Verify that the magnitude of the unit vector equals 1. The magnitude of [latex]-\\frac{5}{13}\\boldsymbol{i}+\\frac{12}{13}\\boldsymbol{j}[\/latex] is given as<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{{\\left(-\\frac{5}{13}\\right)}^{2}+{\\left(\\frac{12}{13}\\right)}^{2}}&=\\sqrt{\\frac{25}{169}+\\frac{144}{169}} \\\\ &=\\sqrt{\\frac{169}{169}}=1 \\end{align}[\/latex]<\/p>\n<p>The vector\u00a0[latex]u=\\frac{5}{13}[\/latex] <strong><em>i<\/em><\/strong> [latex]+\\frac{12}{13}[\/latex] <strong><em>j<\/em><\/strong> is the unit vector in the same direction as <strong><em>v<\/em><\/strong> [latex]=\\langle -5,12\\rangle[\/latex].<\/p>\n<\/div>\n<\/div>\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-4096\">\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":4,"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-4096","chapter","type-chapter","status-publish","hentry"],"part":4126,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4096","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":5,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4096\/revisions"}],"predecessor-version":[{"id":6485,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4096\/revisions\/6485"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/4126"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4096\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=4096"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=4096"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=4096"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=4096"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}