{"id":155,"date":"2014-12-11T02:30:08","date_gmt":"2014-12-11T02:30:08","guid":{"rendered":"https:\/\/courses.candelalearning.com\/colphysics\/?post_type=chapter&#038;p=155"},"modified":"2016-02-17T03:18:57","modified_gmt":"2016-02-17T03:18:57","slug":"2-2-vectors-scalars-and-coordinate-systems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-physics\/chapter\/2-2-vectors-scalars-and-coordinate-systems\/","title":{"raw":"Vectors, Scalars, and Coordinate Systems","rendered":"Vectors, Scalars, and Coordinate Systems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<div class=\"titlepage\">\r\n<div class=\"cnx-gentext-section cnx-gentext-n\">\r\n<div class=\"abstract\">\r\n<div class=\"itemizedlist\">\r\n<ul class=\"itemizedlist\">\r\n\t<li class=\"listitem\">Define and distinguish between scalar and vector quantities.<\/li>\r\n\t<li class=\"listitem\">Assign a coordinate system for a scenario involving one-dimensional motion.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"m42124-import-auto-id1778274\" class=\"figure\" title=\"Figure 2.6.\">\r\n<div class=\"body\">\r\n<div class=\"mediaobject\">\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20101525\/Figure_02_02_00.jpg\" alt=\"A small jet airplane flying toward the left.\" width=\"300\" height=\"428\" \/> Figure 1. The motion of this Eclipse Concept jet can be described in terms of the distance it has traveled (a scalar quantity) or its displacement in a specific direction (a vector quantity). In order to specify the direction of motion, its displacement must be described based on a coordinate system. In this case, it may be convenient to choose motion toward the left as positive motion (it is the forward direction for the plane), although in many cases, the x-coordinate runs from left to right, with motion to the right as positive and motion to the left as negative. (credit: Armchair Aviator, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"title\">What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A <em class=\"glossterm\"> vector<\/em><a id=\"id658381\" class=\"indexterm\"><\/a> is any quantity with both <span class=\"emphasis\"><em>magnitude and direction<\/em><\/span>. Other examples of vectors include a velocity of 90 km\/h east and a force of 500 newtons straight down.<\/div>\r\n<\/div>\r\nThe direction of a vector in one-dimensional motion is given simply by a plus <span class=\"token\"> ( + ) <\/span> or minus <span class=\"token\">(\u2212)<\/span> sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector\u2019s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.\r\n\r\nSome physical quantities, like distance, either have no direction or none is specified. A <em class=\"glossterm\"> scalar<\/em><a id=\"id658725\" class=\"indexterm\"><\/a> is any quantity that has a magnitude, but no direction. For example, a <span class=\"token\">20\u00baC<\/span> temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km\/h speed limit, a person\u2019s 1.8 m height, and a distance of 2.0 m are all scalars\u2014quantities with no specified direction. Note, however, that a scalar can be negative, such as a <span class=\"token\">\u221220\u00baC<\/span> temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.\r\n<div class=\"section\" title=\"Coordinate Systems for One-Dimensional Motion\">\r\n<div class=\"titlepage\">\r\n<div>\r\n<div>\r\n<h2 id=\"m42124-fs-id1655694\"><span class=\"cnx-gentext-section cnx-gentext-t\">Coordinate Systems for One-Dimensional Motion<\/span><\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure 1, it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.\r\n<div id=\"m42124-import-auto-id1758074\" class=\"figure\" title=\"Figure 2.7.\">\r\n<div class=\"body\">\r\n<div class=\"mediaobject\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"200\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20101526\/Figure_02_02_00b.jpg\" alt=\"An x y coordinate system. An arrow pointing toward the right shows the positive x direction. Negative x is toward the left. An arrow pointing up shows the positive y direction. Negative y points downward.\" width=\"200\" height=\"250\" \/> Figure 2. It is usually convenient to consider motion upward or to the right as positive ( + ) and motion downward or to the left as negative (\u2212).[\/caption]\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3 class=\"title\">Check Your Understanding<\/h3>\r\nA person\u2019s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.\r\n<div class=\"title\">\r\n<h4><strong><span class=\"epub-only pre-text\"> Solution<\/span><\/strong><\/h4>\r\n<div class=\"body\">Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Conceptual Questions<\/h3>\r\n<div id=\"fs-id1364975\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\r\n<div id=\"fs-id1770280\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"import-auto-id1730117\">1. A student writes, \"<em data-effect=\"italics\">A bird that is diving for prey has a speed of -10 m\/s<\/em>.\" What is wrong with the student\u2019s statement? What has the student actually described? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1773292\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\r\n<div id=\"fs-id1706742\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"import-auto-id1768462\">2. What is the speed of the bird?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1247502\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\r\n<div id=\"fs-id1777549\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"import-auto-id1655589\">3. Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1548043\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\r\n<div id=\"fs-id1778185\" class=\"problem\" data-type=\"problem\">\r\n<p id=\"import-auto-id1611951\">4. A weather forecast states that the temperature is predicted to be -5\u00baC\u00a0the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\">\r\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\r\n<dl id=\"import-auto-id1493068\" class=\"definition\"><dt>scalar:<\/dt><dd id=\"fs-id1322494\">a quantity that is described by magnitude, but not direction<\/dd><\/dl><dl id=\"import-auto-id1823077\" class=\"definition\"><dt>vector:<\/dt><dd id=\"fs-id2576197\">a quantity that is described by both magnitude and direction<\/dd><\/dl><\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<div class=\"titlepage\">\n<div class=\"cnx-gentext-section cnx-gentext-n\">\n<div class=\"abstract\">\n<div class=\"itemizedlist\">\n<ul class=\"itemizedlist\">\n<li class=\"listitem\">Define and distinguish between scalar and vector quantities.<\/li>\n<li class=\"listitem\">Assign a coordinate system for a scenario involving one-dimensional motion.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"m42124-import-auto-id1778274\" class=\"figure\" title=\"Figure 2.6.\">\n<div class=\"body\">\n<div class=\"mediaobject\">\n<div style=\"width: 310px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20101525\/Figure_02_02_00.jpg\" alt=\"A small jet airplane flying toward the left.\" width=\"300\" height=\"428\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The motion of this Eclipse Concept jet can be described in terms of the distance it has traveled (a scalar quantity) or its displacement in a specific direction (a vector quantity). In order to specify the direction of motion, its displacement must be described based on a coordinate system. In this case, it may be convenient to choose motion toward the left as positive motion (it is the forward direction for the plane), although in many cases, the x-coordinate runs from left to right, with motion to the right as positive and motion to the left as negative. (credit: Armchair Aviator, Flickr)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"title\">What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A <em class=\"glossterm\"> vector<\/em><a id=\"id658381\" class=\"indexterm\"><\/a> is any quantity with both <span class=\"emphasis\"><em>magnitude and direction<\/em><\/span>. Other examples of vectors include a velocity of 90 km\/h east and a force of 500 newtons straight down.<\/div>\n<\/div>\n<p>The direction of a vector in one-dimensional motion is given simply by a plus <span class=\"token\"> ( + ) <\/span> or minus <span class=\"token\">(\u2212)<\/span> sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector\u2019s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.<\/p>\n<p>Some physical quantities, like distance, either have no direction or none is specified. A <em class=\"glossterm\"> scalar<\/em><a id=\"id658725\" class=\"indexterm\"><\/a> is any quantity that has a magnitude, but no direction. For example, a <span class=\"token\">20\u00baC<\/span> temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km\/h speed limit, a person\u2019s 1.8 m height, and a distance of 2.0 m are all scalars\u2014quantities with no specified direction. Note, however, that a scalar can be negative, such as a <span class=\"token\">\u221220\u00baC<\/span> temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.<\/p>\n<div class=\"section\" title=\"Coordinate Systems for One-Dimensional Motion\">\n<div class=\"titlepage\">\n<div>\n<div>\n<h2 id=\"m42124-fs-id1655694\"><span class=\"cnx-gentext-section cnx-gentext-t\">Coordinate Systems for One-Dimensional Motion<\/span><\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure 1, it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.<\/p>\n<div id=\"m42124-import-auto-id1758074\" class=\"figure\" title=\"Figure 2.7.\">\n<div class=\"body\">\n<div class=\"mediaobject\">\n<div style=\"width: 210px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20101526\/Figure_02_02_00b.jpg\" alt=\"An x y coordinate system. An arrow pointing toward the right shows the positive x direction. Negative x is toward the left. An arrow pointing up shows the positive y direction. Negative y points downward.\" width=\"200\" height=\"250\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. It is usually convenient to consider motion upward or to the right as positive ( + ) and motion downward or to the left as negative (\u2212).<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3 class=\"title\">Check Your Understanding<\/h3>\n<p>A person\u2019s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.<\/p>\n<div class=\"title\">\n<h4><strong><span class=\"epub-only pre-text\"> Solution<\/span><\/strong><\/h4>\n<div class=\"body\">Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Conceptual Questions<\/h3>\n<div id=\"fs-id1364975\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\n<div id=\"fs-id1770280\" class=\"problem\" data-type=\"problem\">\n<p id=\"import-auto-id1730117\">1. A student writes, &#8220;<em data-effect=\"italics\">A bird that is diving for prey has a speed of -10 m\/s<\/em>.&#8221; What is wrong with the student\u2019s statement? What has the student actually described? Explain.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1773292\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\n<div id=\"fs-id1706742\" class=\"problem\" data-type=\"problem\">\n<p id=\"import-auto-id1768462\">2. What is the speed of the bird?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1247502\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\n<div id=\"fs-id1777549\" class=\"problem\" data-type=\"problem\">\n<p id=\"import-auto-id1655589\">3. Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1548043\" class=\"exercise\" data-type=\"exercise\" data-element-type=\"conceptual-questions\">\n<div id=\"fs-id1778185\" class=\"problem\" data-type=\"problem\">\n<p id=\"import-auto-id1611951\">4. A weather forecast states that the temperature is predicted to be -5\u00baC\u00a0the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"import-auto-id1493068\" class=\"definition\">\n<dt>scalar:<\/dt>\n<dd id=\"fs-id1322494\">a quantity that is described by magnitude, but not direction<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1823077\" class=\"definition\">\n<dt>vector:<\/dt>\n<dd id=\"fs-id2576197\">a quantity that is described by both magnitude and direction<\/dd>\n<\/dl>\n<\/div>\n<p>&nbsp;<\/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-155\">\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>College Physics. <strong>Authored by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics\">http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Located at License<\/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":5,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Physics\",\"author\":\"OpenStax College\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Located at License\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-155","chapter","type-chapter","status-publish","hentry"],"part":7456,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/155","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/users\/5"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/155\/revisions"}],"predecessor-version":[{"id":11627,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/155\/revisions\/11627"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/parts\/7456"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/155\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/media?parent=155"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapter-type?post=155"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/contributor?post=155"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/license?post=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}