{"id":14172,"date":"2018-06-15T19:22:47","date_gmt":"2018-06-15T19:22:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/key-concepts-glossary-19\/"},"modified":"2018-06-15T19:22:47","modified_gmt":"2018-06-15T19:22:47","slug":"key-concepts-glossary-19","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/key-concepts-glossary-19\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The position vector has its initial point at the origin.<\/li>\n<li>If the position vector is the same for two vectors, they are equal.<\/li>\n<li>Vectors are defined by their magnitude and direction.<\/li>\n<li>If two vectors have the same magnitude and direction, they are equal.<\/li>\n<li>Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements.<\/li>\n<li>Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same.<\/li>\n<li>Vectors are comprised of two components: the horizontal component along the positive <em>x<\/em>-axis, and the vertical component along the positive <em>y<\/em>-axis.<\/li>\n<li>The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.<\/li>\n<li>The magnitude of a vector in the rectangular coordinate system is [latex]|v|=\\sqrt{{a}^{2}+{b}^{2}}[\/latex].<\/li>\n<li>In the rectangular coordinate system, unit vectors may be represented in terms of <strong> [latex]i[\/latex] <\/strong> and <strong> [latex]j[\/latex] <\/strong> where<strong> [latex]i[\/latex] <\/strong>represents the horizontal component and<strong> [latex]j[\/latex] <\/strong>represents the vertical component. Then, <strong><em>v<\/em><\/strong> = a<strong><em>i<\/em><\/strong> + b<strong><em>j<\/em><\/strong>\u2009 is a scalar multiple of<strong> [latex]v[\/latex] <\/strong>by real numbers [latex]a\\text{ and }b[\/latex].<\/li>\n<li>Adding and subtracting vectors in terms of <em>i<\/em> and <em>j<\/em> consists of adding or subtracting corresponding coefficients of <em>i<\/em> and corresponding coefficients of <em>j<\/em>.<\/li>\n<li>A vector <em>v<\/em> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> is written in terms of magnitude and direction as [latex]v=|v|\\cos \\theta i+|v|\\sin \\theta j[\/latex].<\/li>\n<li>The dot product of two vectors is the product of the<strong> [latex]i[\/latex] <\/strong>terms plus the product of the<strong> [latex]j[\/latex] <\/strong>terms.<\/li>\n<li>We can use the dot product to find the angle between two vectors.<\/li>\n<li>Dot products are useful for many types of physics applications.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133447852\" class=\"definition\">\n<dt>dot product<\/dt>\n<dd id=\"fs-id1165133447857\">given two vectors, the sum of the product of the horizontal components and the product of the vertical components<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133447862\" class=\"definition\">\n<dt>initial point<\/dt>\n<dd id=\"fs-id1165135369492\">the origin of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369497\" class=\"definition\">\n<dt>magnitude<\/dt>\n<dd id=\"fs-id1165135369502\">the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369506\" class=\"definition\">\n<dt>resultant<\/dt>\n<dd id=\"fs-id1165135369512\">a vector that results from addition or subtraction of two vectors, or from scalar multiplication<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238406\" class=\"definition\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1165135238411\">a quantity associated with magnitude but not direction; a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238414\" class=\"definition\">\n<dt>scalar multiplication<\/dt>\n<dd id=\"fs-id1165135238419\">the product of a constant and each component of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238423\" class=\"definition\">\n<dt>standard position<\/dt>\n<dd id=\"fs-id1165135369538\">the placement of a vector with the initial point at [latex]\\left(0,0\\right)[\/latex] and the terminal point [latex]\\left(a,b\\right)[\/latex], represented by the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates of the original vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134036770\" class=\"definition\">\n<dt>terminal point<\/dt>\n<dd id=\"fs-id1165133243502\">the end point of a vector, usually represented by an arrow indicating its direction<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133243505\" class=\"definition\">\n<dt>unit vector<\/dt>\n<dd id=\"fs-id1165133243510\">a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the <em>x<\/em>-axis and is defined as [latex]{v}_{1}=\\langle 1,0\\rangle [\/latex] the vertical unit vector runs along the <em>y<\/em>-axis and is defined as [latex]{v}_{2}=\\langle 0,1\\rangle [\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906701\" class=\"definition\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1165131906706\">a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906711\" class=\"definition\">\n<dt>vector addition<\/dt>\n<dd id=\"fs-id1165135700056\">the sum of two vectors, found by adding corresponding components<\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>The position vector has its initial point at the origin.<\/li>\n<li>If the position vector is the same for two vectors, they are equal.<\/li>\n<li>Vectors are defined by their magnitude and direction.<\/li>\n<li>If two vectors have the same magnitude and direction, they are equal.<\/li>\n<li>Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements.<\/li>\n<li>Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same.<\/li>\n<li>Vectors are comprised of two components: the horizontal component along the positive <em>x<\/em>-axis, and the vertical component along the positive <em>y<\/em>-axis.<\/li>\n<li>The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.<\/li>\n<li>The magnitude of a vector in the rectangular coordinate system is [latex]|v|=\\sqrt{{a}^{2}+{b}^{2}}[\/latex].<\/li>\n<li>In the rectangular coordinate system, unit vectors may be represented in terms of <strong> [latex]i[\/latex] <\/strong> and <strong> [latex]j[\/latex] <\/strong> where<strong> [latex]i[\/latex] <\/strong>represents the horizontal component and<strong> [latex]j[\/latex] <\/strong>represents the vertical component. Then, <strong><em>v<\/em><\/strong> = a<strong><em>i<\/em><\/strong> + b<strong><em>j<\/em><\/strong>\u2009 is a scalar multiple of<strong> [latex]v[\/latex] <\/strong>by real numbers [latex]a\\text{ and }b[\/latex].<\/li>\n<li>Adding and subtracting vectors in terms of <em>i<\/em> and <em>j<\/em> consists of adding or subtracting corresponding coefficients of <em>i<\/em> and corresponding coefficients of <em>j<\/em>.<\/li>\n<li>A vector <em>v<\/em> = <em>a<strong>i<\/strong><\/em> + <em>b<strong>j<\/strong><\/em> is written in terms of magnitude and direction as [latex]v=|v|\\cos \\theta i+|v|\\sin \\theta j[\/latex].<\/li>\n<li>The dot product of two vectors is the product of the<strong> [latex]i[\/latex] <\/strong>terms plus the product of the<strong> [latex]j[\/latex] <\/strong>terms.<\/li>\n<li>We can use the dot product to find the angle between two vectors.<\/li>\n<li>Dot products are useful for many types of physics applications.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133447852\" class=\"definition\">\n<dt>dot product<\/dt>\n<dd id=\"fs-id1165133447857\">given two vectors, the sum of the product of the horizontal components and the product of the vertical components<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133447862\" class=\"definition\">\n<dt>initial point<\/dt>\n<dd id=\"fs-id1165135369492\">the origin of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369497\" class=\"definition\">\n<dt>magnitude<\/dt>\n<dd id=\"fs-id1165135369502\">the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135369506\" class=\"definition\">\n<dt>resultant<\/dt>\n<dd id=\"fs-id1165135369512\">a vector that results from addition or subtraction of two vectors, or from scalar multiplication<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238406\" class=\"definition\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id1165135238411\">a quantity associated with magnitude but not direction; a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238414\" class=\"definition\">\n<dt>scalar multiplication<\/dt>\n<dd id=\"fs-id1165135238419\">the product of a constant and each component of a vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135238423\" class=\"definition\">\n<dt>standard position<\/dt>\n<dd id=\"fs-id1165135369538\">the placement of a vector with the initial point at [latex]\\left(0,0\\right)[\/latex] and the terminal point [latex]\\left(a,b\\right)[\/latex], represented by the change in the <em>x<\/em>-coordinates and the change in the <em>y<\/em>-coordinates of the original vector<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134036770\" class=\"definition\">\n<dt>terminal point<\/dt>\n<dd id=\"fs-id1165133243502\">the end point of a vector, usually represented by an arrow indicating its direction<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133243505\" class=\"definition\">\n<dt>unit vector<\/dt>\n<dd id=\"fs-id1165133243510\">a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the <em>x<\/em>-axis and is defined as [latex]{v}_{1}=\\langle 1,0\\rangle[\/latex] the vertical unit vector runs along the <em>y<\/em>-axis and is defined as [latex]{v}_{2}=\\langle 0,1\\rangle[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906701\" class=\"definition\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1165131906706\">a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131906711\" class=\"definition\">\n<dt>vector addition<\/dt>\n<dd id=\"fs-id1165135700056\">the sum of two vectors, found by adding corresponding components<\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\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-14172\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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":23485,"menu_order":6,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-14172","chapter","type-chapter","status-publish","hentry"],"part":14144,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/14172","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/14172\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/parts\/14144"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/14172\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/media?parent=14172"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapter-type?post=14172"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/contributor?post=14172"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/license?post=14172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}