{"id":502,"date":"2025-10-09T15:15:53","date_gmt":"2025-10-09T15:15:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=502"},"modified":"2025-11-18T13:38:02","modified_gmt":"2025-11-18T13:38:02","slug":"three-dimensional-space-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/three-dimensional-space-fresh-take\/","title":{"raw":"Three-Dimensional Space: Fresh Take","rendered":"Three-Dimensional Space: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Describe three-dimensional space mathematically<\/li>\r\n \t<li>Find and graph points in 3D space using their [latex](x, y, z)[\/latex] coordinates<\/li>\r\n \t<li>Write the distance formula in three dimensions<\/li>\r\n \t<li>Write the equations for simple planes and spheres<\/li>\r\n \t<li><span data-sheets-root=\"1\">Perform vector operations in \u211d3<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Three-Dimensional Coordinate Systems<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Real-world problems happen in three dimensions\u2014flight paths, terrain navigation, molecular structures. To handle these, we extend the 2D coordinate system by adding a third axis perpendicular to the other two.<\/p>\r\n<p class=\"whitespace-normal break-words\">The <strong>three-dimensional rectangular coordinate system<\/strong> ([latex]\\mathbb{R}^3[\/latex]) consists of three mutually perpendicular axes: [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex]. Their arrangement follows the <strong>right-hand rule<\/strong>:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Point your right hand's fingers along the positive [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Curl them toward the positive [latex]y[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Your thumb points in the direction of the positive [latex]z[\/latex]-axis<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Plotting Points in 3D<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A point in 3D space is written as [latex](x, y, z)[\/latex]. To plot it: move [latex]x[\/latex] units along the [latex]x[\/latex]-axis, then [latex]y[\/latex] units in the [latex]y[\/latex]-direction, then [latex]z[\/latex] units in the [latex]z[\/latex]-direction. Imagine constructing a rectangular box with the origin and your point as opposite corners.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Tip<\/strong>: 3D visualization on paper is tricky\u2014use physical objects or online 3D graphing tools to build intuition.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Coordinate Planes and Octants<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Three axes create three coordinate planes, each defined by setting one coordinate to zero:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]xy[\/latex]-plane<\/strong>: [latex]z = 0[\/latex] (the \"floor\")<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]xz[\/latex]-plane<\/strong>: [latex]y = 0[\/latex] (one \"wall\")<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]yz[\/latex]-plane<\/strong>: [latex]x = 0[\/latex] (other \"wall\")<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">These three planes divide space into eight regions called <strong>octants<\/strong> (like quadrants, but in 3D).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Distance Formula in 3D<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The distance between points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_2)[\/latex] extends the 2D formula by adding the third coordinate:<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This measures the diagonal of a rectangular box with the two points as opposite corners.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>3D vs 2D: New Possibilities<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Three dimensions allow geometric relationships impossible in 2D:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Skew lines<\/strong>: Lines that don't intersect and aren't parallel (impossible in 2D)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Interconnected circles<\/strong>: Circles linked like chain links without sharing any points (impossible in 2D)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the point [latex](-2, 3, -1)[\/latex] in three-dimensional space.[reveal-answer q=\"173554276\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173554276\"][caption id=\"attachment_3480\" align=\"aligncenter\" width=\"376\"]<img class=\"wp-image-3480 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192236\/2-2-tryitans1.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn. One corner is labeled (-2, 3, -1).\" width=\"376\" height=\"359\" \/> Figure 4.\u00a0Sketch of point [latex](-2, 3, -1)[\/latex][\/caption][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the distance between points [latex]P_1=(1, -5, 4)[\/latex] and\u00a0[latex]P_2=(4, -1, -1)[\/latex].[reveal-answer q=\"267361074\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"267361074\"][latex]5\\sqrt{2}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7713625&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k_EYI3baAvI&amp;video_target=tpm-plugin-w5z1tf7u-k_EYI3baAvI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.12_transcript.html\">\u201cCP 2.12\u201d here (opens in new window)<\/a><\/center><\/section>\r\n<h2 data-type=\"title\">Writing Equations in [latex]\\mathbb{R}^3[\/latex]<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">The same equation describes different objects depending on dimension. For example, [latex]x = 0[\/latex] is a point in 1D, a line in 2D, and a plane in 3D.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Planes Parallel to Coordinate Planes<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">In 3D, setting one coordinate equal to a constant creates a plane:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = a[\/latex]: Plane parallel to [latex]yz[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = b[\/latex]: Plane parallel to [latex]xz[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = c[\/latex]: Plane parallel to [latex]xy[\/latex]-plane<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The constant value tells you how far the plane is from the origin. For example, [latex]z = 5[\/latex] is a horizontal plane [latex]5[\/latex] units above the [latex]xy[\/latex]-plane.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Spheres in 3D<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A sphere with center [latex](a, b, c)[\/latex] and radius [latex]r[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex](x - a)^2 + (y - b)^2 + (z - c)^2 = r^2[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This extends the 2D circle equation by adding [latex](z - c)^2[\/latex]. To find [latex]r[\/latex], use the 3D distance formula from the center to any point on the sphere.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Cylinders and Surfaces<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">When an equation uses only two coordinates (like [latex](x - a)^2 + (y - b)^2 = r^2[\/latex]), the missing coordinate is unrestricted. This creates a cylinder extending infinitely in that direction.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Multiple Objects from One Equation<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Equations like [latex](x - 4)(z - 2) = 0[\/latex] factor into [latex]x = 4[\/latex] or [latex]z = 2[\/latex], describing two intersecting planes.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write an equation of the plane passing through point [latex](1, -6, -4)[\/latex] that is parallel to the [latex]xy[\/latex]-plane.[reveal-answer q=\"287486231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"287486231\"][latex]z=-4[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the standard equation of the sphere with center [latex](-2, 4, 5)[\/latex] containing point [latex](4, 4, -1)[\/latex].[reveal-answer q=\"345141882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345141882\"]<center>[latex](x+2)^2 + (y-4)^2+ (z+5)^2 = 52[\/latex].<\/center>[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the equation of the sphere with diameter [latex]PQ[\/latex], where [latex]P=(2, -1, -3)[\/latex] and\u00a0[latex]Q=(-2, 5, -1)[\/latex].[reveal-answer q=\"926217364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"926217364\"][latex]x^2 + (y-2)^2 + (z+2)^2 = 14[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7713626&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=mXmxE06s5YM&amp;video_target=tpm-plugin-zn4i6xd7-mXmxE06s5YM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.15_transcript.html\">\u201cCP 2.15\u201d here (opens in new window)<\/a><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">Describe the set of points that satisfies [latex](y+2)(z-3)=0[\/latex], and graph the set.[reveal-answer q=\"8472611427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"8472611427\"]The set of points forms the two planes [latex]y=-2[\/latex] and\u00a0[latex]z=3[\/latex].\u00a0<span data-type=\"newline\">\r\n<\/span>[caption id=\"attachment_5052\" align=\"aligncenter\" width=\"733\"]<img class=\"size-full wp-image-5052\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194439\/Figure-2.38.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x z-plane. The second is parallel to the y z-plane at the value of z = 3. They are perpendicular to each other.\" width=\"733\" height=\"586\" \/> Figure 7. The set of points forms the two planes [latex]y=\u22122[\/latex] and [latex]z=3[\/latex].[\/caption][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Describe the set of points in three dimensional space that satisfies [latex]x^{2}+(z-2)^{2}=16[\/latex], and graph the surface.[reveal-answer q=\"682732716\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"682732716\"]A cylinder of radius 4 centered on the line with [latex]x=0[\/latex] and\u00a0[latex]z=2[\/latex].\u00a0<span data-type=\"newline\">\r\n<\/span>[caption id=\"attachment_5054\" align=\"aligncenter\" width=\"523\"]<img class=\"size-full wp-image-5054\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194913\/Try-It-2.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a cylinder parallel to the y-axis and centered around the y-axis.\" width=\"523\" height=\"475\" \/> Figure 9. A cylinder of radius [latex]4[\/latex] centered on the line with [latex]x=0[\/latex] and [latex]z=2[\/latex].[\/caption][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above Try IT.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7713627&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=mLpeNWqn464&amp;video_target=tpm-plugin-fr29097o-mLpeNWqn464\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.17_transcript.html\">\u201cCP 2.17\u201d here (opens in new window)<\/a><\/center><\/section>\r\n<h2>Working with Vectors in Three Dimensions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Three-dimensional vectors work exactly like 2D vectors\u2014they have magnitude and direction. The only difference? They point in 3D space and have three components instead of two.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Component Form in 3D<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A vector [latex]\\mathbf{v} = \\langle x, y, z \\rangle[\/latex] starts at the origin [latex](0, 0, 0)[\/latex] and ends at [latex](x, y, z)[\/latex]. The zero vector is [latex]\\mathbf{0} = \\langle 0, 0, 0 \\rangle[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Vector Operations: Add the Third Component<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\mathbf{v} = \\langle x_1, y_1, z_1 \\rangle[\/latex] and [latex]\\mathbf{w} = \\langle x_2, y_2, z_2 \\rangle[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Addition<\/strong>: [latex]\\mathbf{v} + \\mathbf{w} = \\langle x_1 + x_2, y_1 + y_2, z_1 + z_2 \\rangle[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Scalar multiplication<\/strong>: [latex]k\\mathbf{v} = \\langle kx_1, ky_1, kz_1 \\rangle[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Magnitude<\/strong>: [latex]||\\mathbf{v}|| = \\sqrt{x_1^2 + y_1^2 + z_1^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Unit vector<\/strong>: [latex]\\frac{1}{||\\mathbf{v}||}\\mathbf{v}[\/latex] (normalize by dividing by magnitude)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Standard Unit Vectors: Meet [latex]\\mathbf{k}[\/latex]<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The 2D unit vectors [latex]\\mathbf{i}[\/latex] and [latex]\\mathbf{j}[\/latex] get a third companion:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\mathbf{i} = \\langle 1, 0, 0 \\rangle[\/latex] (along [latex]x[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\mathbf{j} = \\langle 0, 1, 0 \\rangle[\/latex] (along [latex]y[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\mathbf{k} = \\langle 0, 0, 1 \\rangle[\/latex] (along [latex]z[\/latex]-axis)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Any vector can be written: [latex]\\mathbf{v} = x\\mathbf{i} + y\\mathbf{j} + z\\mathbf{k}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Converting from Points<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For a vector from [latex]P = (x_1, y_1, z_1)[\/latex] to [latex]Q = (x_2, y_2, z_2)[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\overrightarrow{PQ} = \\langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \\rangle[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">All 2D vector properties (commutative, associative, distributive) apply in 3D. Geometric methods (parallelogram rule, triangle method) work the same way too.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]S=(3, 8, 2)[\/latex] and [latex]T=(2, -1, 3)[\/latex]. Express [latex]\\overrightarrow{ST}[\/latex] in component form and in standard unit form.[reveal-answer q=\"736898611\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"736898611\"]<center>[latex]\\overrightarrow{ST} = \\langle-1,-9,1 \\rangle = -\\mathbf{i}-9\\mathbf{j}+\\mathbf{k}[\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]\\mathbf{v} = \\langle -1,-1,1 \\rangle[\/latex] and [latex]\\mathbf{w} = \\langle 2,0,1 \\rangle[\/latex]. Find a unit vector in the direction of [latex]5\\mathbf{v}+3\\mathbf{w}[\/latex].[reveal-answer q=\"467325473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"467325473\"][latex]\\langle \\frac{1}{3\\sqrt{10}},-\\frac{5}{3\\sqrt{10}},\\frac{8}{3\\sqrt{10}} \\rangle[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7713628&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=4P7oGFpfAuk&amp;video_target=tpm-plugin-swx82xkd-4P7oGFpfAuk\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.19_transcript.html\">\u201cCP 2.19\u201d here (opens in new window)<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Describe three-dimensional space mathematically<\/li>\n<li>Find and graph points in 3D space using their [latex](x, y, z)[\/latex] coordinates<\/li>\n<li>Write the distance formula in three dimensions<\/li>\n<li>Write the equations for simple planes and spheres<\/li>\n<li><span data-sheets-root=\"1\">Perform vector operations in \u211d3<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Three-Dimensional Coordinate Systems<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Real-world problems happen in three dimensions\u2014flight paths, terrain navigation, molecular structures. To handle these, we extend the 2D coordinate system by adding a third axis perpendicular to the other two.<\/p>\n<p class=\"whitespace-normal break-words\">The <strong>three-dimensional rectangular coordinate system<\/strong> ([latex]\\mathbb{R}^3[\/latex]) consists of three mutually perpendicular axes: [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex]. Their arrangement follows the <strong>right-hand rule<\/strong>:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Point your right hand&#8217;s fingers along the positive [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Curl them toward the positive [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Your thumb points in the direction of the positive [latex]z[\/latex]-axis<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Plotting Points in 3D<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A point in 3D space is written as [latex](x, y, z)[\/latex]. To plot it: move [latex]x[\/latex] units along the [latex]x[\/latex]-axis, then [latex]y[\/latex] units in the [latex]y[\/latex]-direction, then [latex]z[\/latex] units in the [latex]z[\/latex]-direction. Imagine constructing a rectangular box with the origin and your point as opposite corners.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Tip<\/strong>: 3D visualization on paper is tricky\u2014use physical objects or online 3D graphing tools to build intuition.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Coordinate Planes and Octants<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Three axes create three coordinate planes, each defined by setting one coordinate to zero:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>[latex]xy[\/latex]-plane<\/strong>: [latex]z = 0[\/latex] (the &#8220;floor&#8221;)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>[latex]xz[\/latex]-plane<\/strong>: [latex]y = 0[\/latex] (one &#8220;wall&#8221;)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>[latex]yz[\/latex]-plane<\/strong>: [latex]x = 0[\/latex] (other &#8220;wall&#8221;)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">These three planes divide space into eight regions called <strong>octants<\/strong> (like quadrants, but in 3D).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Distance Formula in 3D<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The distance between points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_2)[\/latex] extends the 2D formula by adding the third coordinate:<\/p>\n<p class=\"whitespace-normal break-words\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This measures the diagonal of a rectangular box with the two points as opposite corners.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>3D vs 2D: New Possibilities<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Three dimensions allow geometric relationships impossible in 2D:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Skew lines<\/strong>: Lines that don&#8217;t intersect and aren&#8217;t parallel (impossible in 2D)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Interconnected circles<\/strong>: Circles linked like chain links without sharing any points (impossible in 2D)<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the point [latex](-2, 3, -1)[\/latex] in three-dimensional space.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q173554276\">Show Solution<\/button><\/p>\n<div id=\"q173554276\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_3480\" aria-describedby=\"caption-attachment-3480\" style=\"width: 376px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3480 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192236\/2-2-tryitans1.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn. One corner is labeled (-2, 3, -1).\" width=\"376\" height=\"359\" \/><figcaption id=\"caption-attachment-3480\" class=\"wp-caption-text\">Figure 4.\u00a0Sketch of point [latex](-2, 3, -1)[\/latex]<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the distance between points [latex]P_1=(1, -5, 4)[\/latex] and\u00a0[latex]P_2=(4, -1, -1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q267361074\">Show Solution<\/button><\/p>\n<div id=\"q267361074\" class=\"hidden-answer\" style=\"display: none\">[latex]5\\sqrt{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7713625&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k_EYI3baAvI&amp;video_target=tpm-plugin-w5z1tf7u-k_EYI3baAvI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.12_transcript.html\">\u201cCP 2.12\u201d here (opens in new window)<\/a><\/div>\n<\/section>\n<h2 data-type=\"title\">Writing Equations in [latex]\\mathbb{R}^3[\/latex]<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The same equation describes different objects depending on dimension. For example, [latex]x = 0[\/latex] is a point in 1D, a line in 2D, and a plane in 3D.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Planes Parallel to Coordinate Planes<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">In 3D, setting one coordinate equal to a constant creates a plane:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x = a[\/latex]: Plane parallel to [latex]yz[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = b[\/latex]: Plane parallel to [latex]xz[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = c[\/latex]: Plane parallel to [latex]xy[\/latex]-plane<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The constant value tells you how far the plane is from the origin. For example, [latex]z = 5[\/latex] is a horizontal plane [latex]5[\/latex] units above the [latex]xy[\/latex]-plane.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Spheres in 3D<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A sphere with center [latex](a, b, c)[\/latex] and radius [latex]r[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex](x - a)^2 + (y - b)^2 + (z - c)^2 = r^2[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This extends the 2D circle equation by adding [latex](z - c)^2[\/latex]. To find [latex]r[\/latex], use the 3D distance formula from the center to any point on the sphere.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Cylinders and Surfaces<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When an equation uses only two coordinates (like [latex](x - a)^2 + (y - b)^2 = r^2[\/latex]), the missing coordinate is unrestricted. This creates a cylinder extending infinitely in that direction.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Multiple Objects from One Equation<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Equations like [latex](x - 4)(z - 2) = 0[\/latex] factor into [latex]x = 4[\/latex] or [latex]z = 2[\/latex], describing two intersecting planes.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write an equation of the plane passing through point [latex](1, -6, -4)[\/latex] that is parallel to the [latex]xy[\/latex]-plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q287486231\">Show Solution<\/button><\/p>\n<div id=\"q287486231\" class=\"hidden-answer\" style=\"display: none\">[latex]z=-4[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the standard equation of the sphere with center [latex](-2, 4, 5)[\/latex] containing point [latex](4, 4, -1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q345141882\">Show Solution<\/button><\/p>\n<div id=\"q345141882\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex](x+2)^2 + (y-4)^2+ (z+5)^2 = 52[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the equation of the sphere with diameter [latex]PQ[\/latex], where [latex]P=(2, -1, -3)[\/latex] and\u00a0[latex]Q=(-2, 5, -1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q926217364\">Show Solution<\/button><\/p>\n<div id=\"q926217364\" class=\"hidden-answer\" style=\"display: none\">[latex]x^2 + (y-2)^2 + (z+2)^2 = 14[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7713626&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=mXmxE06s5YM&amp;video_target=tpm-plugin-zn4i6xd7-mXmxE06s5YM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.15_transcript.html\">\u201cCP 2.15\u201d here (opens in new window)<\/a><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Describe the set of points that satisfies [latex](y+2)(z-3)=0[\/latex], and graph the set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8472611427\">Show Solution<\/button><\/p>\n<div id=\"q8472611427\" class=\"hidden-answer\" style=\"display: none\">The set of points forms the two planes [latex]y=-2[\/latex] and\u00a0[latex]z=3[\/latex].\u00a0<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"attachment_5052\" aria-describedby=\"caption-attachment-5052\" style=\"width: 733px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5052\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194439\/Figure-2.38.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x z-plane. The second is parallel to the y z-plane at the value of z = 3. They are perpendicular to each other.\" width=\"733\" height=\"586\" \/><figcaption id=\"caption-attachment-5052\" class=\"wp-caption-text\">Figure 7. The set of points forms the two planes [latex]y=\u22122[\/latex] and [latex]z=3[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Describe the set of points in three dimensional space that satisfies [latex]x^{2}+(z-2)^{2}=16[\/latex], and graph the surface.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q682732716\">Show Solution<\/button><\/p>\n<div id=\"q682732716\" class=\"hidden-answer\" style=\"display: none\">A cylinder of radius 4 centered on the line with [latex]x=0[\/latex] and\u00a0[latex]z=2[\/latex].\u00a0<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"attachment_5054\" aria-describedby=\"caption-attachment-5054\" style=\"width: 523px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5054\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194913\/Try-It-2.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a cylinder parallel to the y-axis and centered around the y-axis.\" width=\"523\" height=\"475\" \/><figcaption id=\"caption-attachment-5054\" class=\"wp-caption-text\">Figure 9. A cylinder of radius [latex]4[\/latex] centered on the line with [latex]x=0[\/latex] and [latex]z=2[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above Try IT.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7713627&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=mLpeNWqn464&amp;video_target=tpm-plugin-fr29097o-mLpeNWqn464\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.17_transcript.html\">\u201cCP 2.17\u201d here (opens in new window)<\/a><\/div>\n<\/section>\n<h2>Working with Vectors in Three Dimensions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Three-dimensional vectors work exactly like 2D vectors\u2014they have magnitude and direction. The only difference? They point in 3D space and have three components instead of two.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Component Form in 3D<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A vector [latex]\\mathbf{v} = \\langle x, y, z \\rangle[\/latex] starts at the origin [latex](0, 0, 0)[\/latex] and ends at [latex](x, y, z)[\/latex]. The zero vector is [latex]\\mathbf{0} = \\langle 0, 0, 0 \\rangle[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Vector Operations: Add the Third Component<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\mathbf{v} = \\langle x_1, y_1, z_1 \\rangle[\/latex] and [latex]\\mathbf{w} = \\langle x_2, y_2, z_2 \\rangle[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Addition<\/strong>: [latex]\\mathbf{v} + \\mathbf{w} = \\langle x_1 + x_2, y_1 + y_2, z_1 + z_2 \\rangle[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Scalar multiplication<\/strong>: [latex]k\\mathbf{v} = \\langle kx_1, ky_1, kz_1 \\rangle[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Magnitude<\/strong>: [latex]||\\mathbf{v}|| = \\sqrt{x_1^2 + y_1^2 + z_1^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Unit vector<\/strong>: [latex]\\frac{1}{||\\mathbf{v}||}\\mathbf{v}[\/latex] (normalize by dividing by magnitude)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Standard Unit Vectors: Meet [latex]\\mathbf{k}[\/latex]<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The 2D unit vectors [latex]\\mathbf{i}[\/latex] and [latex]\\mathbf{j}[\/latex] get a third companion:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\mathbf{i} = \\langle 1, 0, 0 \\rangle[\/latex] (along [latex]x[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\mathbf{j} = \\langle 0, 1, 0 \\rangle[\/latex] (along [latex]y[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\mathbf{k} = \\langle 0, 0, 1 \\rangle[\/latex] (along [latex]z[\/latex]-axis)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Any vector can be written: [latex]\\mathbf{v} = x\\mathbf{i} + y\\mathbf{j} + z\\mathbf{k}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Converting from Points<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For a vector from [latex]P = (x_1, y_1, z_1)[\/latex] to [latex]Q = (x_2, y_2, z_2)[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\overrightarrow{PQ} = \\langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \\rangle[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">All 2D vector properties (commutative, associative, distributive) apply in 3D. Geometric methods (parallelogram rule, triangle method) work the same way too.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]S=(3, 8, 2)[\/latex] and [latex]T=(2, -1, 3)[\/latex]. Express [latex]\\overrightarrow{ST}[\/latex] in component form and in standard unit form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q736898611\">Show Solution<\/button><\/p>\n<div id=\"q736898611\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\overrightarrow{ST} = \\langle-1,-9,1 \\rangle = -\\mathbf{i}-9\\mathbf{j}+\\mathbf{k}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]\\mathbf{v} = \\langle -1,-1,1 \\rangle[\/latex] and [latex]\\mathbf{w} = \\langle 2,0,1 \\rangle[\/latex]. Find a unit vector in the direction of [latex]5\\mathbf{v}+3\\mathbf{w}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q467325473\">Show Solution<\/button><\/p>\n<div id=\"q467325473\" class=\"hidden-answer\" style=\"display: none\">[latex]\\langle \\frac{1}{3\\sqrt{10}},-\\frac{5}{3\\sqrt{10}},\\frac{8}{3\\sqrt{10}} \\rangle[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7713628&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=4P7oGFpfAuk&amp;video_target=tpm-plugin-swx82xkd-4P7oGFpfAuk\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.19_transcript.html\">\u201cCP 2.19\u201d here (opens in new window)<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":15,"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-502","chapter","type-chapter","status-publish","hentry"],"part":420,"module-header":"fresh_take","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\/502","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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/502\/revisions"}],"predecessor-version":[{"id":2597,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/502\/revisions\/2597"}],"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\/502\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=502"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=502"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=502"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=502"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}