{"id":490,"date":"2025-10-09T15:14:32","date_gmt":"2025-10-09T15:14:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=490"},"modified":"2026-08-25T15:12:36","modified_gmt":"2026-08-25T15:12:36","slug":"vector-basics-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-basics-learn-it-1\/","title":{"raw":"Vector Basics: Learn It 1","rendered":"Vector Basics: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Explain what vectors are and how to write them using proper notation, with real-world examples of vector quantities<\/li>\r\n \t<li>Perform basic vector operations (scalar multiplication, addition, subtraction)<\/li>\r\n \t<li>Write vectors using their components and find how long they are (their magnitude)<\/li>\r\n \t<li>Express a vector in terms of unit vectors<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Vector Basics<\/h2>\r\n<p class=\"whitespace-normal break-words\">In mathematics, some quantities can be completely described by a single number\u2014like temperature, mass, or distance. These are called scalars. But other quantities need more information to be fully understood.<\/p>\r\n<p class=\"whitespace-normal break-words\">Consider describing movement or force. If you say \"the wind is blowing at 15 mph,\" that's incomplete information. You also need to know the direction\u2014north, southeast, etc. Similarly, when describing a force, you need both its strength and the direction it's applied.<\/p>\r\n<p class=\"whitespace-normal break-words\">A quantity that has both <strong>magnitude<\/strong> (size) and <strong>direction<\/strong> is called a <strong>vector<\/strong>. We denote vectors using boldface letters like [latex]\\mathbf{v}[\/latex].<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>vector<\/h3>\r\nA vector is a quantity that has both magnitude and direction.\r\n\r\n<\/section>\r\n<h3>Vector Representation<\/h3>\r\n<p class=\"whitespace-normal break-words\">You can represent a vector in a plane using a directed line segment (an arrow). The starting point is the <strong>initial point<\/strong> and the ending point is the <strong>terminal point<\/strong>. The arrow shows the vector's direction, while the length of the line segment represents its <strong>magnitude<\/strong>. We write [latex]||\\mathbf{v}||[\/latex] to denote the magnitude of vector [latex]\\mathbf{v}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">The <strong>zero vector<\/strong>, denoted [latex]\\mathbf{0}[\/latex], has its initial point and terminal point at the same location. This is the only vector without a specific direction\u2014by convention, we can assign it any direction that's convenient for the problem.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Equivalent vectors<\/strong> have the same magnitude and direction. We treat equivalent vectors as equal, even when they start at different points. If vectors [latex]\\mathbf{v}[\/latex] and [latex]\\mathbf{w}[\/latex] are equivalent, we write:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = \\mathbf{w}[\/latex]<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>equivalent vectors<\/h3>\r\nVectors are said to be <strong>equivalent vectors<\/strong> if they have the same magnitude and direction.\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">The arrows in Figure 1(b) show equivalent vectors\u2014each has identical length and direction.<\/p>\r\n\r\n[caption id=\"attachment_673\" align=\"aligncenter\" width=\"622\"]<img class=\"wp-image-673 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17180011\/2-1-1.jpeg\" alt=\"This figure has two images. The first is labeled \u201ca\u201d and has a line segment representing vector v. The line segment begins at the initial point and goes to the terminal point. There is an arrowhead at the terminal point. The second image is labeled \u201cb\u201d and is five vectors, each labeled v sub 1, v sub 2, v sub 3, v sub 4, v sub 5. They all are pointing in the same direction and have the same length.\" width=\"622\" height=\"247\" \/> Figure 1 (a) A vector is represented by a directed line segment from its initial point to its terminal point. (b) Vectors\u00a0[latex]{\\bf{v}}_{1}[\/latex] through\u00a0[latex]{\\bf{v}}_{5}[\/latex] are equivalent.[\/caption]\r\n<p class=\"whitespace-normal break-words\">Two vectors are <strong>parallel<\/strong> if they point in the same direction or exact opposite directions. We'll explore this concept in more detail later.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Are [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] equivalent vectors?\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\bf{v}[\/latex] has initial point [latex](3,2)[\/latex] and terminal point [latex](7,2)[\/latex]\r\n\r\n[latex]\\bf{w}[\/latex] has initial point [latex](1,\u22124)[\/latex] and terminal point [latex](1,0)[\/latex]<\/li>\r\n \t<li>[latex]\\bf{v}[\/latex] has initial point [latex](0,0)[\/latex] and terminal point [latex](1,1) [\/latex]\r\n\r\n[latex]\\bf{w}[\/latex] has initial point [latex](\u22122,2)[\/latex] and terminal point [latex](\u22121,3)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1167795055165\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167795055165\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>The vectors are each [latex]4[\/latex] units long, but they are oriented in different directions. So [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] are not equivalent (Figure 2.10).\r\n\r\n[caption id=\"attachment_690\" align=\"aligncenter\" width=\"342\"]<img class=\"wp-image-690 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17190445\/2-1-10.jpeg\" alt=\"This figure is a Cartesian coordinate system with two vectors. The first vector labeled \u201cv\u201d has initial point at (3, 2) and terminal point (7, 2). It is parallel to the x-axis. The second vector is labeled \u201cw\u201d and has initial point (1, -4) and terminal point (1, 0). It is parallel to the y-axis.\" width=\"342\" height=\"384\" \/> Figure 1. These vectors are not equivalent.[\/caption]<\/li>\r\n \t<li>Based on Figure 2.11, and using a bit of geometry, it is clear these vectors have the same length and the same direction, so [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] are equivalent.<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_691\" align=\"aligncenter\" width=\"267\"]<img class=\"wp-image-691 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17190539\/2-1-11.jpeg\" alt=\"This figure is a Cartesian coordinate system with two vectors. The first vector labeled \u201cv\u201d has initial point at (0, 0) and terminal point (1, 1). The second vector is labeled \u201cw\u201d and has initial point (-2, 2) and terminal point (-1, 3).\" width=\"267\" height=\"272\" \/> Figure 2. These vectors are equivalent.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Vector Notation<\/strong>\r\n<ul style=\"list-style-type: disc;\">\r\n \t<li>In print, we use boldface letters like [latex]\\mathbf{v}[\/latex].<\/li>\r\n \t<li>When writing by hand, draw an arrow over the letter: [latex]\\overrightarrow{v}[\/latex].<\/li>\r\n \t<li>When a vector goes from point [latex]P[\/latex] to point [latex]Q[\/latex], you can write [latex]\\overrightarrow{PQ}[\/latex] to show both direction and location.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Sketch a vector in the plane from initial point [latex]P(1,1)[\/latex] to terminal point [latex]Q(8,5)[\/latex].\r\n<div id=\"fs-id1167793361764\" class=\"exercise\">[reveal-answer q=\"fs-id1167794055154\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794055154\"]\r\nSee Figure 2. Because the vector goes from point [latex]P[\/latex] to point [latex]Q[\/latex], we name it [latex]\\overrightarrow{PQ}[\/latex].[caption id=\"attachment_677\" align=\"aligncenter\" width=\"417\"]<img class=\"wp-image-677 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17180848\/2-1-2.jpeg\" alt=\"This figure is a graph of the first quadrant. There is a line segment beginning at the ordered pair (1, 1). Also, this point is labeled \u201cP.\u201d The line segment ends at the ordered pair (8, 5) and is labeled \u201cQ.\u201d The line segment is labeled \u201cPQ.\u201d\" width=\"417\" height=\"309\" \/> Figure 2. The vector with initial point (1,1) and terminal point (8,5) is named [latex]\\overrightarrow{PQ}[\/latex].[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]328428[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Explain what vectors are and how to write them using proper notation, with real-world examples of vector quantities<\/li>\n<li>Perform basic vector operations (scalar multiplication, addition, subtraction)<\/li>\n<li>Write vectors using their components and find how long they are (their magnitude)<\/li>\n<li>Express a vector in terms of unit vectors<\/li>\n<\/ul>\n<\/section>\n<h2>Vector Basics<\/h2>\n<p class=\"whitespace-normal break-words\">In mathematics, some quantities can be completely described by a single number\u2014like temperature, mass, or distance. These are called scalars. But other quantities need more information to be fully understood.<\/p>\n<p class=\"whitespace-normal break-words\">Consider describing movement or force. If you say &#8220;the wind is blowing at 15 mph,&#8221; that&#8217;s incomplete information. You also need to know the direction\u2014north, southeast, etc. Similarly, when describing a force, you need both its strength and the direction it&#8217;s applied.<\/p>\n<p class=\"whitespace-normal break-words\">A quantity that has both <strong>magnitude<\/strong> (size) and <strong>direction<\/strong> is called a <strong>vector<\/strong>. We denote vectors using boldface letters like [latex]\\mathbf{v}[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>vector<\/h3>\n<p>A vector is a quantity that has both magnitude and direction.<\/p>\n<\/section>\n<h3>Vector Representation<\/h3>\n<p class=\"whitespace-normal break-words\">You can represent a vector in a plane using a directed line segment (an arrow). The starting point is the <strong>initial point<\/strong> and the ending point is the <strong>terminal point<\/strong>. The arrow shows the vector&#8217;s direction, while the length of the line segment represents its <strong>magnitude<\/strong>. We write [latex]||\\mathbf{v}||[\/latex] to denote the magnitude of vector [latex]\\mathbf{v}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">The <strong>zero vector<\/strong>, denoted [latex]\\mathbf{0}[\/latex], has its initial point and terminal point at the same location. This is the only vector without a specific direction\u2014by convention, we can assign it any direction that&#8217;s convenient for the problem.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Equivalent vectors<\/strong> have the same magnitude and direction. We treat equivalent vectors as equal, even when they start at different points. If vectors [latex]\\mathbf{v}[\/latex] and [latex]\\mathbf{w}[\/latex] are equivalent, we write:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\mathbf{v} = \\mathbf{w}[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>equivalent vectors<\/h3>\n<p>Vectors are said to be <strong>equivalent vectors<\/strong> if they have the same magnitude and direction.<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">The arrows in Figure 1(b) show equivalent vectors\u2014each has identical length and direction.<\/p>\n<figure id=\"attachment_673\" aria-describedby=\"caption-attachment-673\" style=\"width: 622px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-673 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17180011\/2-1-1.jpeg\" alt=\"This figure has two images. The first is labeled \u201ca\u201d and has a line segment representing vector v. The line segment begins at the initial point and goes to the terminal point. There is an arrowhead at the terminal point. The second image is labeled \u201cb\u201d and is five vectors, each labeled v sub 1, v sub 2, v sub 3, v sub 4, v sub 5. They all are pointing in the same direction and have the same length.\" width=\"622\" height=\"247\" \/><figcaption id=\"caption-attachment-673\" class=\"wp-caption-text\">Figure 1 (a) A vector is represented by a directed line segment from its initial point to its terminal point. (b) Vectors\u00a0[latex]{\\bf{v}}_{1}[\/latex] through\u00a0[latex]{\\bf{v}}_{5}[\/latex] are equivalent.<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">Two vectors are <strong>parallel<\/strong> if they point in the same direction or exact opposite directions. We&#8217;ll explore this concept in more detail later.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Are [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] equivalent vectors?<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\bf{v}[\/latex] has initial point [latex](3,2)[\/latex] and terminal point [latex](7,2)[\/latex]\n<p>[latex]\\bf{w}[\/latex] has initial point [latex](1,\u22124)[\/latex] and terminal point [latex](1,0)[\/latex]<\/li>\n<li>[latex]\\bf{v}[\/latex] has initial point [latex](0,0)[\/latex] and terminal point [latex](1,1)[\/latex]\n<p>[latex]\\bf{w}[\/latex] has initial point [latex](\u22122,2)[\/latex] and terminal point [latex](\u22121,3)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167795055165\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167795055165\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>The vectors are each [latex]4[\/latex] units long, but they are oriented in different directions. So [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] are not equivalent (Figure 2.10).<br \/>\n<figure id=\"attachment_690\" aria-describedby=\"caption-attachment-690\" style=\"width: 342px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-690 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17190445\/2-1-10.jpeg\" alt=\"This figure is a Cartesian coordinate system with two vectors. The first vector labeled \u201cv\u201d has initial point at (3, 2) and terminal point (7, 2). It is parallel to the x-axis. The second vector is labeled \u201cw\u201d and has initial point (1, -4) and terminal point (1, 0). It is parallel to the y-axis.\" width=\"342\" height=\"384\" \/><figcaption id=\"caption-attachment-690\" class=\"wp-caption-text\">Figure 1. These vectors are not equivalent.<\/figcaption><\/figure>\n<\/li>\n<li>Based on Figure 2.11, and using a bit of geometry, it is clear these vectors have the same length and the same direction, so [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] are equivalent.<\/li>\n<\/ol>\n<figure id=\"attachment_691\" aria-describedby=\"caption-attachment-691\" style=\"width: 267px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-691 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17190539\/2-1-11.jpeg\" alt=\"This figure is a Cartesian coordinate system with two vectors. The first vector labeled \u201cv\u201d has initial point at (0, 0) and terminal point (1, 1). The second vector is labeled \u201cw\u201d and has initial point (-2, 2) and terminal point (-1, 3).\" width=\"267\" height=\"272\" \/><figcaption id=\"caption-attachment-691\" class=\"wp-caption-text\">Figure 2. These vectors are equivalent.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Vector Notation<\/strong><\/p>\n<ul style=\"list-style-type: disc;\">\n<li>In print, we use boldface letters like [latex]\\mathbf{v}[\/latex].<\/li>\n<li>When writing by hand, draw an arrow over the letter: [latex]\\overrightarrow{v}[\/latex].<\/li>\n<li>When a vector goes from point [latex]P[\/latex] to point [latex]Q[\/latex], you can write [latex]\\overrightarrow{PQ}[\/latex] to show both direction and location.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a vector in the plane from initial point [latex]P(1,1)[\/latex] to terminal point [latex]Q(8,5)[\/latex].<\/p>\n<div id=\"fs-id1167793361764\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794055154\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794055154\" class=\"hidden-answer\" style=\"display: none\">\nSee Figure 2. Because the vector goes from point [latex]P[\/latex] to point [latex]Q[\/latex], we name it [latex]\\overrightarrow{PQ}[\/latex].<\/p>\n<figure id=\"attachment_677\" aria-describedby=\"caption-attachment-677\" style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-677 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/17180848\/2-1-2.jpeg\" alt=\"This figure is a graph of the first quadrant. There is a line segment beginning at the ordered pair (1, 1). Also, this point is labeled \u201cP.\u201d The line segment ends at the ordered pair (8, 5) and is labeled \u201cQ.\u201d The line segment is labeled \u201cPQ.\u201d\" width=\"417\" height=\"309\" \/><figcaption id=\"caption-attachment-677\" class=\"wp-caption-text\">Figure 2. The vector with initial point (1,1) and terminal point (8,5) is named [latex]\\overrightarrow{PQ}[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm328428\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=328428&theme=lumen&iframe_resize_id=ohm328428&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"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-490","chapter","type-chapter","status-publish","hentry"],"part":420,"module-header":"learn_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\/490","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":11,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/490\/revisions"}],"predecessor-version":[{"id":2901,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/490\/revisions\/2901"}],"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\/490\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=490"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=490"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=490"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=490"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}