{"id":229,"date":"2025-02-13T22:45:19","date_gmt":"2025-02-13T22:45:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/vectors\/"},"modified":"2025-08-15T14:54:46","modified_gmt":"2025-08-15T14:54:46","slug":"vectors","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/vectors\/","title":{"raw":"Vectors: Learn It 1","rendered":"Vectors: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>View vectors geometrically.<\/li>\r\n \t<li>Find magnitude and direction.<\/li>\r\n \t<li>Find the component form of a vector.<\/li>\r\n \t<li>Find the unit vector in the direction of v.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>A Geometric View of Vectors<\/h2>\r\nA <strong>vector<\/strong> is a specific quantity drawn as a line segment with an arrowhead at one end. It has an <strong>initial point<\/strong>, where it begins, and a <strong>terminal point<\/strong>, where it ends. A vector is defined by its <strong>magnitude<\/strong>, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment.\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">There are various symbols that distinguish vectors from other quantities:\r\n<ul>\r\n \t<li>Lower case, boldfaced type, with or without an arrow on top such as [latex]\\boldsymbol{v,u,w,\\stackrel{\\to }{v},\\stackrel{\\to }{u},\\stackrel{\\to }{w}}[\/latex].<\/li>\r\n \t<li>Given initial point [latex]P[\/latex] and terminal point [latex]Q[\/latex], a vector can be represented as [latex]\\stackrel{\\to }{PQ}[\/latex]. The arrowhead on top is what indicates that it is not just a line, but a directed line segment.<\/li>\r\n \t<li>Given an initial point of [latex]\\left(0,0\\right)[\/latex] and terminal point [latex]\\left(a,b\\right)[\/latex], a vector may be represented as [latex]\\langle a,b\\rangle [\/latex].<\/li>\r\n<\/ul>\r\n<\/section>This last symbol [latex]\\langle a,b\\rangle [\/latex] has special significance. It is called the <strong>standard position<\/strong>. The <strong>position vector<\/strong> has an initial point [latex]\\left(0,0\\right)[\/latex] and a terminal point [latex]\\langle a,b\\rangle [\/latex]. To change any vector into the position vector, we think about the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates. Thus, if the initial point of a vector [latex]\\stackrel{\\to }{CD}[\/latex] is [latex]C\\left({x}_{1},{y}_{1}\\right)[\/latex] and the terminal point is [latex]D\\left({x}_{2},{y}_{2}\\right)[\/latex], then the position vector is found by calculating\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\stackrel{\\to }{AB}&amp;=\\langle {x}_{2}-{x}_{1},{y}_{2}-{y}_{1}\\rangle \\\\ &amp;=\\langle a,b\\rangle \\end{align}[\/latex]<\/div>\r\nThe original vector [latex]\\stackrel{\\to }{CD}[\/latex] and the position vector [latex]\\stackrel{\\to }{AB}[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181129\/CNX_Precalc_Figure_08_08_0032.jpg\" alt=\"Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.\" width=\"487\" height=\"290\" \/>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>vector<\/h3>\r\nA vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at [latex]\\left(0,0\\right)[\/latex] and is identified by its terminal point [latex]\\langle a,b\\rangle [\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Consider the vector whose initial point is [latex]P\\left(2,3\\right)[\/latex] and terminal point is [latex]Q\\left(6,4\\right)[\/latex]. Find the position vector.[reveal-answer q=\"454005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"454005\"]The position vector is found by subtracting one <em>x<\/em>-coordinate from the other <em>x<\/em>-coordinate, and one <em>y<\/em>-coordinate from the other <em>y<\/em>-coordinate. Thus\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{v}&amp;=\\langle 6 - 2,4 - 3\\rangle \\\\ &amp;=\\langle 4,1\\rangle \\end{align}[\/latex]<\/p>\r\nThe position vector begins at [latex]\\left(0,0\\right)[\/latex] and terminates at [latex]\\left(4,1\\right)[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181132\/CNX_Precalc_Figure_08_08_0222.jpg\" alt=\"Plot of the original vector in blue and the position vector in orange extending from the origin.\" width=\"487\" height=\"349\" \/>\r\n\r\nWe see that the position vector is [latex]\\langle 4,1\\rangle [\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the position vector given that vector<em><strong> v <\/strong><\/em>has an initial point at [latex]\\left(-3,2\\right)[\/latex] and a terminal point at [latex]\\left(4,5\\right)[\/latex], then graph both vectors in the same plane.[reveal-answer q=\"823850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"823850\"]The position vector is found using the following calculation:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{v}&amp;=\\langle 4-\\left(-3\\right),5 - 2\\rangle \\\\ &amp;=\\langle 7,3\\rangle \\end{align}[\/latex]<\/p>\r\nThus, the position vector begins at [latex]\\left(0,0\\right)[\/latex] and terminates at [latex]\\left(7,3\\right)[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181134\/CNX_Precalc_Figure_08_08_004n2.jpg\" alt=\"Plot of the two given vectors their same position vector.\" width=\"487\" height=\"328\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Draw a vector [latex]\\boldsymbol{v}[\/latex] that connects from the origin to the point [latex]\\left(3,5\\right)[\/latex].[reveal-answer q=\"884783\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"884783\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181235\/CNX_Precalc_Figure_08_08_0062.jpg\" alt=\"A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.\" \/>[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]173913[\/ohm_question]<\/section>\r\n<dl id=\"fs-id1165131906711\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135700056\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>View vectors geometrically.<\/li>\n<li>Find magnitude and direction.<\/li>\n<li>Find the component form of a vector.<\/li>\n<li>Find the unit vector in the direction of v.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>A Geometric View of Vectors<\/h2>\n<p>A <strong>vector<\/strong> is a specific quantity drawn as a line segment with an arrowhead at one end. It has an <strong>initial point<\/strong>, where it begins, and a <strong>terminal point<\/strong>, where it ends. A vector is defined by its <strong>magnitude<\/strong>, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">There are various symbols that distinguish vectors from other quantities:<\/p>\n<ul>\n<li>Lower case, boldfaced type, with or without an arrow on top such as [latex]\\boldsymbol{v,u,w,\\stackrel{\\to }{v},\\stackrel{\\to }{u},\\stackrel{\\to }{w}}[\/latex].<\/li>\n<li>Given initial point [latex]P[\/latex] and terminal point [latex]Q[\/latex], a vector can be represented as [latex]\\stackrel{\\to }{PQ}[\/latex]. The arrowhead on top is what indicates that it is not just a line, but a directed line segment.<\/li>\n<li>Given an initial point of [latex]\\left(0,0\\right)[\/latex] and terminal point [latex]\\left(a,b\\right)[\/latex], a vector may be represented as [latex]\\langle a,b\\rangle[\/latex].<\/li>\n<\/ul>\n<\/section>\n<p>This last symbol [latex]\\langle a,b\\rangle[\/latex] has special significance. It is called the <strong>standard position<\/strong>. The <strong>position vector<\/strong> has an initial point [latex]\\left(0,0\\right)[\/latex] and a terminal point [latex]\\langle a,b\\rangle[\/latex]. To change any vector into the position vector, we think about the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates. Thus, if the initial point of a vector [latex]\\stackrel{\\to }{CD}[\/latex] is [latex]C\\left({x}_{1},{y}_{1}\\right)[\/latex] and the terminal point is [latex]D\\left({x}_{2},{y}_{2}\\right)[\/latex], then the position vector is found by calculating<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\stackrel{\\to }{AB}&=\\langle {x}_{2}-{x}_{1},{y}_{2}-{y}_{1}\\rangle \\\\ &=\\langle a,b\\rangle \\end{align}[\/latex]<\/div>\n<p>The original vector [latex]\\stackrel{\\to }{CD}[\/latex] and the position vector [latex]\\stackrel{\\to }{AB}[\/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\/27181129\/CNX_Precalc_Figure_08_08_0032.jpg\" alt=\"Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.\" width=\"487\" height=\"290\" \/><\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>vector<\/h3>\n<p>A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at [latex]\\left(0,0\\right)[\/latex] and is identified by its terminal point [latex]\\langle a,b\\rangle[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Consider the vector whose initial point is [latex]P\\left(2,3\\right)[\/latex] and terminal point is [latex]Q\\left(6,4\\right)[\/latex]. Find the position vector.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q454005\">Show Solution<\/button><\/p>\n<div id=\"q454005\" class=\"hidden-answer\" style=\"display: none\">The position vector is found by subtracting one <em>x<\/em>-coordinate from the other <em>x<\/em>-coordinate, and one <em>y<\/em>-coordinate from the other <em>y<\/em>-coordinate. Thus<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{v}&=\\langle 6 - 2,4 - 3\\rangle \\\\ &=\\langle 4,1\\rangle \\end{align}[\/latex]<\/p>\n<p>The position vector begins at [latex]\\left(0,0\\right)[\/latex] and terminates at [latex]\\left(4,1\\right)[\/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\/27181132\/CNX_Precalc_Figure_08_08_0222.jpg\" alt=\"Plot of the original vector in blue and the position vector in orange extending from the origin.\" width=\"487\" height=\"349\" \/><\/p>\n<p>We see that the position vector is [latex]\\langle 4,1\\rangle[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the position vector given that vector<em><strong> v <\/strong><\/em>has an initial point at [latex]\\left(-3,2\\right)[\/latex] and a terminal point at [latex]\\left(4,5\\right)[\/latex], then graph both vectors in the same plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q823850\">Show Solution<\/button><\/p>\n<div id=\"q823850\" class=\"hidden-answer\" style=\"display: none\">The position vector is found using the following calculation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\boldsymbol{v}&=\\langle 4-\\left(-3\\right),5 - 2\\rangle \\\\ &=\\langle 7,3\\rangle \\end{align}[\/latex]<\/p>\n<p>Thus, the position vector begins at [latex]\\left(0,0\\right)[\/latex] and terminates at [latex]\\left(7,3\\right)[\/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\/27181134\/CNX_Precalc_Figure_08_08_004n2.jpg\" alt=\"Plot of the two given vectors their same position vector.\" width=\"487\" height=\"328\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Draw a vector [latex]\\boldsymbol{v}[\/latex] that connects from the origin to the point [latex]\\left(3,5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q884783\">Show Solution<\/button><\/p>\n<div id=\"q884783\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181235\/CNX_Precalc_Figure_08_08_0062.jpg\" alt=\"A vector from the origin to (3,5) - a line with an arrow at the (3,5) endpoint.\" \/><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm173913\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173913&theme=lumen&iframe_resize_id=ohm173913&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<dl id=\"fs-id1165131906711\" class=\"definition\">\n<dd id=\"fs-id1165135700056\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/229"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/229\/revisions"}],"predecessor-version":[{"id":2966,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/229\/revisions\/2966"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/229\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=229"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=229"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=229"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}