{"id":5442,"date":"2022-06-02T18:18:26","date_gmt":"2022-06-02T18:18:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=5442"},"modified":"2022-10-21T00:01:47","modified_gmt":"2022-10-21T00:01:47","slug":"writing-equations-in-three-dimensions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/writing-equations-in-three-dimensions\/","title":{"raw":"Writing Equations in Three Dimensions","rendered":"Writing Equations in Three Dimensions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Write the distance formula in three dimensions.<\/span><\/li>\r\n \t<li><span class=\"os-abstract-content\">Write the equations for simple planes and spheres.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Writing Equations in [latex]\\mathbb{R}^3[\/latex]<\/h2>\r\n<p id=\"fs-id1163724077303\">Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in [latex]\\mathbb{R}^3[\/latex]. First, we start with a simple equation. Compare the graphs of the equation\u00a0[latex]x=0[\/latex] in\u00a0[latex]\\mathbb{R}[\/latex],\u00a0[latex]\\mathbb{R}^2[\/latex], and\u00a0[latex]\\mathbb{R}^3[\/latex] (Figure 1). From these graphs, we can see the same equation can describe a point, a line, or a plane.<\/p>\r\n\r\n[caption id=\"attachment_5039\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-5039\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26191900\/Figure-2.32.jpg\" alt=\"This figure has three images. The first is a horizontal axis with a point drawn at 0. The second is the two dimensional Cartesian coordinate plane. The third is the 3-dimensional coordinate system. It is inside of a box and has a grid drawn at the y z-plane.\" width=\"975\" height=\"386\" \/> Figure 1. (a) In [latex]\u211d[\/latex], the equation [latex]x=0[\/latex] describes a single point. (b) In [latex]\u211d^2[\/latex], the equation [latex]x=0[\/latex] describes a line, the [latex]y[\/latex]-axis. (c) In [latex]\u211d^3[\/latex], the equation [latex]x=0[\/latex] describes a plane, the [latex]yz[\/latex]-plane.[\/caption]In space, the equation [latex]x=0[\/latex] describes all points [latex](0, y, z)[\/latex]. This equation defines the [latex]yz[\/latex]-plane. Similarly, the [latex]xy[\/latex]-plane contains all points of the form [latex](x, y, 0)[\/latex]. The equation [latex]z=0[\/latex] defines the [latex]xy[\/latex]-plane and the equation [latex]y=0[\/latex] describes the [latex]xz[\/latex]-plane (Figure 2).\r\n\r\n[caption id=\"attachment_5044\" align=\"aligncenter\" width=\"732\"]<img class=\"size-full wp-image-5044\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26192745\/Figure-2.33.jpg\" alt=\"This figure has two images. The first is the 3-dimensional coordinate system. It is inside of a box and has a grid drawn at the x y-plane. The second is the 3-dimensional coordinate system. It is inside of a box and has a grid drawn at the x z-plane.\" width=\"732\" height=\"404\" \/> Figure 2 (a) In space, the equation [latex]z=0[\/latex] describes the [latex]xy[\/latex]-plane. (b) All points in the [latex]xz[\/latex]-plane satisfy the equation [latex]y=0[\/latex].[\/caption]Understanding the equations of the coordinate planes allows us to write an equation for any plane that is parallel to one of the coordinate planes. When a plane is parallel to the [latex]xy[\/latex]-plane, for example, the [latex]z[\/latex]-coordinate of each point in the plane has the same constant value. Only the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates of points in that plane vary from point to point.\r\n\r\n&nbsp;\r\n<div id=\"CNX_Calc_Figure_12_02_008\" class=\"os-figure\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">rule: equations of planes parallel to coordinate planes<\/h3>\r\n\r\n<hr \/>\r\n\r\n<ol id=\"fs-id1163724075425\" type=\"1\">\r\n \t<li>The plane in space that is parallel to the [latex]xy[\/latex]-plane and contains point [latex](a, b, c)[\/latex] can be represented by the equation [latex]z=c[\/latex].<\/li>\r\n \t<li>The plane in space that is parallel to the [latex]xz[\/latex]-plane and contains point [latex](a, b, c)[\/latex] can be represented by the equation [latex]y=b[\/latex].<\/li>\r\n \t<li>The plane in space that is parallel to the [latex]yz[\/latex]-plane and contains point [latex](a, b, c)[\/latex] can be represented by the equation [latex]x=a[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: writing equations of planes parallel to coordinate planes<\/h3>\r\n<ol id=\"fs-id1163723756014\" type=\"a\">\r\n \t<li>Write an equation of the plane passing through point [latex](3, 11, 7)[\/latex] that is parallel to the [latex]yz[\/latex]-plane.<\/li>\r\n \t<li>Find an equation of the plane passing through points [latex](6, -2, 9)[\/latex], [latex](0, -2, 4)[\/latex], and [latex](1, -2, -3)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"485274093\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485274093\"]\r\n<ol id=\"fs-id1163723755666\" type=\"a\">\r\n \t<li>When a plane is parallel to the [latex]yz[\/latex]-plane, only the [latex]y[\/latex]- and [latex]z[\/latex]-coordinates may vary. The [latex]x[\/latex]-coordinate has the same constant value for all points in this plane, so this plane can be represented by the equation [latex]x=3[\/latex].<\/li>\r\n \t<li>Each of the points [latex](6, -2, 9)[\/latex], [latex](0, -2, 4)[\/latex], and [latex](1, -2, -3)[\/latex] has the same\u00a0<em data-effect=\"italics\">y<\/em>-coordinate. This plane can be represented by the equation [latex]y=-2[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nWrite an equation of the plane passing through point [latex](1, -6, -4)[\/latex] that is parallel to the [latex]xy[\/latex]-plane.\r\n\r\n[reveal-answer q=\"287486231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"287486231\"]\r\n\r\n[latex]z=-4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]242489[\/ohm_question]\r\n\r\n<\/div>\r\nAs we have seen, in [latex]\\mathbb{R}^2[\/latex] the equation [latex]x=5[\/latex] describes the vertical line passing through point [latex](5, 0)[\/latex]. This line is parallel to the [latex]y[\/latex]-axis. In a natural extension, the equation [latex]x=5[\/latex] in [latex]\\mathbb{R}^3[\/latex] describes the plane passing through point [latex](5, 0, 0)[\/latex], which is parallel to the [latex]yz[\/latex]-plane. Another natural extension of a familiar equation is found in the equation of a sphere.\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nA\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term68\" data-type=\"term\">sphere<\/span><\/strong>\u00a0is the set of all points in space equidistant from a fixed point, the center of the sphere (Figure 3), just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the\u00a0<em data-effect=\"italics\">radius<\/em>.\r\n\r\n<\/div>\r\n[caption id=\"attachment_5045\" align=\"aligncenter\" width=\"284\"]<img class=\"size-full wp-image-5045\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26192959\/Figure-2.34.jpg\" alt=\"This image is a sphere. It has center at (a, b, c) and has a radius represented with a broken line from the center point (a, b, c) to the edge of the sphere at (x, y, z). The radius is labeled \u201cr.\u201d\" width=\"284\" height=\"252\" \/> Figure 3. Each point [latex](x,y,z)[\/latex] on the surface of a sphere is [latex]r[\/latex] units away from the center [latex](a,b,c)[\/latex].[\/caption]The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">rule: equation of a sphere<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1163723798377\">The sphere with center [latex](a, b, c)[\/latex] and radius [latex]r[\/latex] can be represented by the equation<\/p>\r\n\r\n<center>[latex](x - a)^2 +(y - b)^2+ (z - c)^2 =r^2[\/latex]<\/center>.\r\n<p id=\"fs-id1163723739312\">This equation is known as the\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term69\" data-type=\"term\">standard equation of a sphere<\/span><\/strong>.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding the equation of a sphere<\/h3>\r\n<p id=\"fs-id1163724072839\">Find the standard equation of the sphere with center [latex](10, 7, 4)[\/latex] and point [latex](-1, 3, -2)[\/latex] as shown in\u00a0Figure 4.<\/p>\r\n\r\n[caption id=\"attachment_5046\" align=\"aligncenter\" width=\"473\"]<img class=\"size-full wp-image-5046\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26193144\/Figure-2.35.jpg\" alt=\"This figure is a sphere centered on the point (10, 7, 4) of a 3-dimensional coordinate system. It has radius equal to the square root of 173 and passes through the point (-1, 3, -2).\" width=\"473\" height=\"480\" \/> Figure 4. The sphere centered at [latex](10,7,4)[\/latex] containing point [latex](\u22121,3,\u22122)[\/latex].[\/caption][reveal-answer q=\"942317736\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"942317736\"]\r\n<p id=\"fs-id1163723681381\">Use the distance formula to find the radius [latex]r[\/latex] of the sphere:<\/p>\r\n\r\n<center>[latex]\r\n\\begin{align*}\r\nr &amp;= \\sqrt{(-1 - 10)^2 + (3-7)^2 + (-2 - 4)^2}\\\\\r\n&amp;= \\sqrt{(-11)^2 + (-4)^2 + (-6)^2}\\\\\r\n&amp;= \\sqrt{173}\r\n\\end{align*}\r\n[\/latex].<\/center>\r\n<p id=\"fs-id1163723754519\">The standard equation of the sphere is<\/p>\r\n\r\n<center>[latex](x-10)^2 + (y-7)^2+ (z-4)^2 = 173[\/latex].<\/center>[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nFind the standard equation of the sphere with center [latex](-2, 4, 5)[\/latex] containing point [latex](4, 4, -1)[\/latex].\r\n\r\n[reveal-answer q=\"345141882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345141882\"]\r\n\r\n<center>[latex](x+2)^2 + (y-4)^2+ (z+5)^2 = 52[\/latex].<\/center>[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding the equation of a sphere<\/h3>\r\nLet [latex]P=(-5, 2, 3)[\/latex] and [latex]Q=(3, 4, -1)[\/latex], and suppose line segment [latex]PQ[\/latex] forms the diameter of a sphere (Figure 5). Find the equation of the sphere.\r\n\r\n[caption id=\"attachment_5047\" align=\"aligncenter\" width=\"550\"]<img class=\"size-full wp-image-5047\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26193402\/Figure-2.36.jpg\" alt=\"This figure is the 3-dimensional coordinate system. There are two points labeled. The first point is P = (-5, 2, 3). The second point is Q = (3, 4, -1). There is a line segment drawn between the two points.\" width=\"550\" height=\"471\" \/> Figure 5. Line segment [latex]PQ[\/latex].[\/caption][reveal-answer q=\"177538250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"177538250\"]\r\n<p id=\"fs-id1163723753932\">Since [latex]PQ[\/latex] is a diameter of the sphere, we know the center of the sphere is the midpoint of [latex]PQ[\/latex]. Then,<\/p>\r\n\r\n<center>[latex]\r\n\\begin{align*}\r\nC &amp;= (\\frac{-5+3}{2}, \\frac{2+4}{2}, \\frac{3+(-1)}{2})\\\\\r\n&amp;= (-1, 3, 1)\\\\\r\n\\end{align*}\r\n[\/latex].<\/center>\r\n<p id=\"fs-id1163723934860\">Furthermore, we know the radius of the sphere is half the length of the diameter. This gives<\/p>\r\n\r\n<center>[latex]\r\n\\begin{align*}\r\nr &amp;= \\frac{1}{2}\\sqrt{(-5 - 3)^2 + (2-4)^2 + (3 - (-1))^2}\\\\\r\n&amp;= \\frac{1}{2}\\sqrt{64+4+16}\\\\\r\n&amp;= \\sqrt{21}\r\n\\end{align*}\r\n[\/latex].<\/center>.\r\n<p id=\"fs-id1163724058200\">Then, the equation of the sphere is [latex](x+1)^2 + (y-3)^2 + (z-1)^2 = 21[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nFind 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].\r\n\r\n[reveal-answer q=\"926217364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"926217364\"]\r\n\r\n[latex]x^2 + (y-2)^2 + (z+2)^2 = 14[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try IT.\r\n\r\n<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\"><\/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>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: graphing other equations in three dimensions<\/h3>\r\nDescribe the set of points that satisfies [latex](x-4)(z-2)=0[\/latex], and graph the set.\r\n\r\n[reveal-answer q=\"887239643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"887239643\"]\r\n\r\nWe must have either [latex]x-4=0[\/latex] or [latex]z-2=0[\/latex], so the set of points forms the two planes [latex]x=4[\/latex] and [latex]z=2[\/latex] (Figure 6).\r\n\r\n[caption id=\"attachment_5051\" align=\"aligncenter\" width=\"660\"]<img class=\"size-full wp-image-5051\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194234\/Figure-2.37.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x y-plane. The second is the y z-plane. They are perpendicular to each other.\" width=\"660\" height=\"604\" \/> Figure 6. The set of points satisfying [latex](x\u22124)(z\u22122)=0[\/latex] forms the two planes [latex]x=4[\/latex] and [latex]z=2[\/latex].[\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nDescribe the set of points that satisfies [latex](y+2)(z-3)=0[\/latex], and graph the set.\r\n\r\n[reveal-answer q=\"8472611427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"8472611427\"]\r\n\r\nThe 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>\r\n\r\n[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]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: graphing other equations in three dimensions<\/h3>\r\nDescribe the set of points in three-dimensional space that satisfies [latex](x+2)^{2}+(y-1)^{2}=4[\/latex], and graph the set.\r\n\r\n[reveal-answer q=\"767152709\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767152709\"]\r\n\r\nThe [latex]x[\/latex]- and [latex]y[\/latex]-coordinates form a circle in the [latex]xy[\/latex]-plane of radius 2, centered at [latex](2, 1)[\/latex]. Since there is no restriction on the [latex]z[\/latex]-coordinate, the three-dimensional result is a circular cylinder of radius 2 centered on the line with [latex]x=2[\/latex] and [latex]x=1[\/latex]. The cylinder extends indefinitely in the [latex]z[\/latex]-direction (Figure 8).\r\n\r\n[caption id=\"attachment_5053\" align=\"aligncenter\" width=\"501\"]<img class=\"size-full wp-image-5053\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194649\/Figure-2.381.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a vertical cylinder parallel to the z-axis and centered around line parallel to the z-axis with x = 2 and y = 1.\" width=\"501\" height=\"623\" \/> Figure 8. The set of points satisfying [latex](x\u22122)^2+(y\u22121)^2=4[\/latex]. This is a cylinder of radius [latex]2[\/latex] centered on the line with [latex]x=2[\/latex] and [latex]y=1[\/latex].[\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nDescribe the set of points in three dimensional space that satisfies [latex]x^{2}+(z-2)^{2}=16[\/latex], and graph the surface.\r\n\r\n[reveal-answer q=\"682732716\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"682732716\"]\r\n\r\nA 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>\r\n\r\n[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]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try IT.\r\n\r\n<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\"><\/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><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Write the distance formula in three dimensions.<\/span><\/li>\n<li><span class=\"os-abstract-content\">Write the equations for simple planes and spheres.<\/span><\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Writing Equations in [latex]\\mathbb{R}^3[\/latex]<\/h2>\n<p id=\"fs-id1163724077303\">Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in [latex]\\mathbb{R}^3[\/latex]. First, we start with a simple equation. Compare the graphs of the equation\u00a0[latex]x=0[\/latex] in\u00a0[latex]\\mathbb{R}[\/latex],\u00a0[latex]\\mathbb{R}^2[\/latex], and\u00a0[latex]\\mathbb{R}^3[\/latex] (Figure 1). From these graphs, we can see the same equation can describe a point, a line, or a plane.<\/p>\n<div id=\"attachment_5039\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5039\" class=\"size-full wp-image-5039\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26191900\/Figure-2.32.jpg\" alt=\"This figure has three images. The first is a horizontal axis with a point drawn at 0. The second is the two dimensional Cartesian coordinate plane. The third is the 3-dimensional coordinate system. It is inside of a box and has a grid drawn at the y z-plane.\" width=\"975\" height=\"386\" \/><\/p>\n<p id=\"caption-attachment-5039\" class=\"wp-caption-text\">Figure 1. (a) In [latex]\u211d[\/latex], the equation [latex]x=0[\/latex] describes a single point. (b) In [latex]\u211d^2[\/latex], the equation [latex]x=0[\/latex] describes a line, the [latex]y[\/latex]-axis. (c) In [latex]\u211d^3[\/latex], the equation [latex]x=0[\/latex] describes a plane, the [latex]yz[\/latex]-plane.<\/p>\n<\/div>\n<p>In space, the equation [latex]x=0[\/latex] describes all points [latex](0, y, z)[\/latex]. This equation defines the [latex]yz[\/latex]-plane. Similarly, the [latex]xy[\/latex]-plane contains all points of the form [latex](x, y, 0)[\/latex]. The equation [latex]z=0[\/latex] defines the [latex]xy[\/latex]-plane and the equation [latex]y=0[\/latex] describes the [latex]xz[\/latex]-plane (Figure 2).<\/p>\n<div id=\"attachment_5044\" style=\"width: 742px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5044\" class=\"size-full wp-image-5044\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26192745\/Figure-2.33.jpg\" alt=\"This figure has two images. The first is the 3-dimensional coordinate system. It is inside of a box and has a grid drawn at the x y-plane. The second is the 3-dimensional coordinate system. It is inside of a box and has a grid drawn at the x z-plane.\" width=\"732\" height=\"404\" \/><\/p>\n<p id=\"caption-attachment-5044\" class=\"wp-caption-text\">Figure 2 (a) In space, the equation [latex]z=0[\/latex] describes the [latex]xy[\/latex]-plane. (b) All points in the [latex]xz[\/latex]-plane satisfy the equation [latex]y=0[\/latex].<\/p>\n<\/div>\n<p>Understanding the equations of the coordinate planes allows us to write an equation for any plane that is parallel to one of the coordinate planes. When a plane is parallel to the [latex]xy[\/latex]-plane, for example, the [latex]z[\/latex]-coordinate of each point in the plane has the same constant value. Only the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates of points in that plane vary from point to point.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"CNX_Calc_Figure_12_02_008\" class=\"os-figure\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">rule: equations of planes parallel to coordinate planes<\/h3>\n<hr \/>\n<ol id=\"fs-id1163724075425\" type=\"1\">\n<li>The plane in space that is parallel to the [latex]xy[\/latex]-plane and contains point [latex](a, b, c)[\/latex] can be represented by the equation [latex]z=c[\/latex].<\/li>\n<li>The plane in space that is parallel to the [latex]xz[\/latex]-plane and contains point [latex](a, b, c)[\/latex] can be represented by the equation [latex]y=b[\/latex].<\/li>\n<li>The plane in space that is parallel to the [latex]yz[\/latex]-plane and contains point [latex](a, b, c)[\/latex] can be represented by the equation [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: writing equations of planes parallel to coordinate planes<\/h3>\n<ol id=\"fs-id1163723756014\" type=\"a\">\n<li>Write an equation of the plane passing through point [latex](3, 11, 7)[\/latex] that is parallel to the [latex]yz[\/latex]-plane.<\/li>\n<li>Find an equation of the plane passing through points [latex](6, -2, 9)[\/latex], [latex](0, -2, 4)[\/latex], and [latex](1, -2, -3)[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q485274093\">Show Solution<\/span><\/p>\n<div id=\"q485274093\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1163723755666\" type=\"a\">\n<li>When a plane is parallel to the [latex]yz[\/latex]-plane, only the [latex]y[\/latex]&#8211; and [latex]z[\/latex]-coordinates may vary. The [latex]x[\/latex]-coordinate has the same constant value for all points in this plane, so this plane can be represented by the equation [latex]x=3[\/latex].<\/li>\n<li>Each of the points [latex](6, -2, 9)[\/latex], [latex](0, -2, 4)[\/latex], and [latex](1, -2, -3)[\/latex] has the same\u00a0<em data-effect=\"italics\">y<\/em>-coordinate. This plane can be represented by the equation [latex]y=-2[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q287486231\">Show Solution<\/span><\/p>\n<div id=\"q287486231\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]z=-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm242489\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=242489&theme=oea&iframe_resize_id=ohm242489&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>As we have seen, in [latex]\\mathbb{R}^2[\/latex] the equation [latex]x=5[\/latex] describes the vertical line passing through point [latex](5, 0)[\/latex]. This line is parallel to the [latex]y[\/latex]-axis. In a natural extension, the equation [latex]x=5[\/latex] in [latex]\\mathbb{R}^3[\/latex] describes the plane passing through point [latex](5, 0, 0)[\/latex], which is parallel to the [latex]yz[\/latex]-plane. Another natural extension of a familiar equation is found in the equation of a sphere.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p>A\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term68\" data-type=\"term\">sphere<\/span><\/strong>\u00a0is the set of all points in space equidistant from a fixed point, the center of the sphere (Figure 3), just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the\u00a0<em data-effect=\"italics\">radius<\/em>.<\/p>\n<\/div>\n<div id=\"attachment_5045\" style=\"width: 294px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5045\" class=\"size-full wp-image-5045\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26192959\/Figure-2.34.jpg\" alt=\"This image is a sphere. It has center at (a, b, c) and has a radius represented with a broken line from the center point (a, b, c) to the edge of the sphere at (x, y, z). The radius is labeled \u201cr.\u201d\" width=\"284\" height=\"252\" \/><\/p>\n<p id=\"caption-attachment-5045\" class=\"wp-caption-text\">Figure 3. Each point [latex](x,y,z)[\/latex] on the surface of a sphere is [latex]r[\/latex] units away from the center [latex](a,b,c)[\/latex].<\/p>\n<\/div>\n<p>The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">rule: equation of a sphere<\/h3>\n<hr \/>\n<p id=\"fs-id1163723798377\">The sphere with center [latex](a, b, c)[\/latex] and radius [latex]r[\/latex] can be represented by the equation<\/p>\n<div style=\"text-align: center;\">[latex](x - a)^2 +(y - b)^2+ (z - c)^2 =r^2[\/latex]<\/div>\n<p>.<\/p>\n<p id=\"fs-id1163723739312\">This equation is known as the\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term69\" data-type=\"term\">standard equation of a sphere<\/span><\/strong>.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: finding the equation of a sphere<\/h3>\n<p id=\"fs-id1163724072839\">Find the standard equation of the sphere with center [latex](10, 7, 4)[\/latex] and point [latex](-1, 3, -2)[\/latex] as shown in\u00a0Figure 4.<\/p>\n<div id=\"attachment_5046\" style=\"width: 483px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5046\" class=\"size-full wp-image-5046\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26193144\/Figure-2.35.jpg\" alt=\"This figure is a sphere centered on the point (10, 7, 4) of a 3-dimensional coordinate system. It has radius equal to the square root of 173 and passes through the point (-1, 3, -2).\" width=\"473\" height=\"480\" \/><\/p>\n<p id=\"caption-attachment-5046\" class=\"wp-caption-text\">Figure 4. The sphere centered at [latex](10,7,4)[\/latex] containing point [latex](\u22121,3,\u22122)[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q942317736\">Show Solution<\/span><\/p>\n<div id=\"q942317736\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163723681381\">Use the distance formula to find the radius [latex]r[\/latex] of the sphere:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*}  r &= \\sqrt{(-1 - 10)^2 + (3-7)^2 + (-2 - 4)^2}\\\\  &= \\sqrt{(-11)^2 + (-4)^2 + (-6)^2}\\\\  &= \\sqrt{173}  \\end{align*}[\/latex].<\/div>\n<p id=\"fs-id1163723754519\">The standard equation of the sphere is<\/p>\n<div style=\"text-align: center;\">[latex](x-10)^2 + (y-7)^2+ (z-4)^2 = 173[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q345141882\">Show Solution<\/span><\/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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: finding the equation of a sphere<\/h3>\n<p>Let [latex]P=(-5, 2, 3)[\/latex] and [latex]Q=(3, 4, -1)[\/latex], and suppose line segment [latex]PQ[\/latex] forms the diameter of a sphere (Figure 5). Find the equation of the sphere.<\/p>\n<div id=\"attachment_5047\" style=\"width: 560px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5047\" class=\"size-full wp-image-5047\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26193402\/Figure-2.36.jpg\" alt=\"This figure is the 3-dimensional coordinate system. There are two points labeled. The first point is P = (-5, 2, 3). The second point is Q = (3, 4, -1). There is a line segment drawn between the two points.\" width=\"550\" height=\"471\" \/><\/p>\n<p id=\"caption-attachment-5047\" class=\"wp-caption-text\">Figure 5. Line segment [latex]PQ[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q177538250\">Show Solution<\/span><\/p>\n<div id=\"q177538250\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163723753932\">Since [latex]PQ[\/latex] is a diameter of the sphere, we know the center of the sphere is the midpoint of [latex]PQ[\/latex]. Then,<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*}  C &= (\\frac{-5+3}{2}, \\frac{2+4}{2}, \\frac{3+(-1)}{2})\\\\  &= (-1, 3, 1)\\\\  \\end{align*}[\/latex].<\/div>\n<p id=\"fs-id1163723934860\">Furthermore, we know the radius of the sphere is half the length of the diameter. This gives<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*}  r &= \\frac{1}{2}\\sqrt{(-5 - 3)^2 + (2-4)^2 + (3 - (-1))^2}\\\\  &= \\frac{1}{2}\\sqrt{64+4+16}\\\\  &= \\sqrt{21}  \\end{align*}[\/latex].<\/div>\n<p>.<\/p>\n<p id=\"fs-id1163724058200\">Then, the equation of the sphere is [latex](x+1)^2 + (y-3)^2 + (z-1)^2 = 21[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q926217364\">Show Solution<\/span><\/p>\n<div id=\"q926217364\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x^2 + (y-2)^2 + (z+2)^2 = 14[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>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=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\"><\/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<div class=\"textbox exercises\">\n<h3>Example: graphing other equations in three dimensions<\/h3>\n<p>Describe the set of points that satisfies [latex](x-4)(z-2)=0[\/latex], and graph the set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q887239643\">Show Solution<\/span><\/p>\n<div id=\"q887239643\" class=\"hidden-answer\" style=\"display: none\">\n<p>We must have either [latex]x-4=0[\/latex] or [latex]z-2=0[\/latex], so the set of points forms the two planes [latex]x=4[\/latex] and [latex]z=2[\/latex] (Figure 6).<\/p>\n<div id=\"attachment_5051\" style=\"width: 670px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5051\" class=\"size-full wp-image-5051\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194234\/Figure-2.37.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has two intersecting planes drawn. The first is the x y-plane. The second is the y z-plane. They are perpendicular to each other.\" width=\"660\" height=\"604\" \/><\/p>\n<p id=\"caption-attachment-5051\" class=\"wp-caption-text\">Figure 6. The set of points satisfying [latex](x\u22124)(z\u22122)=0[\/latex] forms the two planes [latex]x=4[\/latex] and [latex]z=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8472611427\">Show Solution<\/span><\/p>\n<div id=\"q8472611427\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<div id=\"attachment_5052\" style=\"width: 743px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5052\" 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\" \/><\/p>\n<p 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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: graphing other equations in three dimensions<\/h3>\n<p>Describe the set of points in three-dimensional space that satisfies [latex](x+2)^{2}+(y-1)^{2}=4[\/latex], and graph the set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q767152709\">Show Solution<\/span><\/p>\n<div id=\"q767152709\" class=\"hidden-answer\" style=\"display: none\">\n<p>The [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates form a circle in the [latex]xy[\/latex]-plane of radius 2, centered at [latex](2, 1)[\/latex]. Since there is no restriction on the [latex]z[\/latex]-coordinate, the three-dimensional result is a circular cylinder of radius 2 centered on the line with [latex]x=2[\/latex] and [latex]x=1[\/latex]. The cylinder extends indefinitely in the [latex]z[\/latex]-direction (Figure 8).<\/p>\n<div id=\"attachment_5053\" style=\"width: 511px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5053\" class=\"size-full wp-image-5053\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26194649\/Figure-2.381.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a vertical cylinder parallel to the z-axis and centered around line parallel to the z-axis with x = 2 and y = 1.\" width=\"501\" height=\"623\" \/><\/p>\n<p id=\"caption-attachment-5053\" class=\"wp-caption-text\">Figure 8. The set of points satisfying [latex](x\u22122)^2+(y\u22121)^2=4[\/latex]. This is a cylinder of radius [latex]2[\/latex] centered on the line with [latex]x=2[\/latex] and [latex]y=1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q682732716\">Show Solution<\/span><\/p>\n<div id=\"q682732716\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<div id=\"attachment_5054\" style=\"width: 533px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5054\" 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\" \/><\/p>\n<p 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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>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\"><\/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<\/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-5442\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 2.17. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>CP 2.15. <strong>Authored by<\/strong>: Ryan Melton. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/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":349141,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at 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