{"id":875,"date":"2025-10-14T15:09:42","date_gmt":"2025-10-14T15:09:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=875"},"modified":"2026-08-25T15:33:23","modified_gmt":"2026-08-25T15:33:23","slug":"three-dimensional-space-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/three-dimensional-space-learn-it-2\/","title":{"raw":"Three-Dimensional Space: Learn It 2","rendered":"Three-Dimensional Space: Learn It 2"},"content":{"raw":"<h2 data-type=\"title\">Writing Equations in [latex]\\mathbb{R}^3[\/latex]<\/h2>\r\n<p class=\"whitespace-normal break-words\">Now that you can represent points in space and calculate distances between them, you can learn to write equations for geometric objects like lines, planes, and curved surfaces in three-dimensional space.<\/p>\r\n<p class=\"whitespace-normal break-words\">The same equation can represent completely different geometric objects depending on the dimensional space you're working in. Consider the simple equation [latex]x = 0[\/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>In [latex]\\mathbb{R}[\/latex] (1D)<\/strong>: Describes a single point<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>In [latex]\\mathbb{R}^2[\/latex] (2D)<\/strong>: Describes a line (the [latex]y[\/latex]-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>In [latex]\\mathbb{R}^3[\/latex] (3D)<\/strong>: Describes a plane (the [latex]yz[\/latex]-plane)<\/li>\r\n<\/ul>\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]\r\n<p class=\"whitespace-normal break-words\">In 3D space, each coordinate plane has a simple equation:<\/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]yz[\/latex]-plane<\/strong>: [latex]x = 0[\/latex] (contains all points [latex](0, y, z)[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]xy[\/latex]-plane<\/strong>: [latex]z = 0[\/latex] (contains all points [latex](x, y, 0)[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]xz[\/latex]-plane<\/strong>: [latex]y = 0[\/latex] (contains all points [latex](x, 0, z)[\/latex])<\/li>\r\n<\/ul>\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]\r\n<p class=\"whitespace-normal break-words\">Once you understand coordinate plane equations, you can write equations for any plane parallel to a coordinate plane. The key insight: when a plane is parallel to a coordinate plane, one coordinate remains constant for all points in that plane.<\/p>\r\n<p class=\"whitespace-normal break-words\">For example, if a plane is parallel to the [latex]xy[\/latex]-plane, every point in that plane has the same [latex]z[\/latex]-coordinate. Only the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates vary from point to point.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>equations of planes parallel to coordinate planes<\/h3>\r\n<ol id=\"fs-id1163724075425\" type=\"1\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parallel to [latex]xy[\/latex]-plane<\/strong>: [latex]z = c[\/latex] (contains point [latex](a, b, c)[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parallel to [latex]xz[\/latex]-plane<\/strong>: [latex]y = b[\/latex] (contains point [latex](a, b, c)[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Parallel to [latex]yz[\/latex]-plane<\/strong>: [latex]x = a[\/latex] (contains point [latex](a, b, c)[\/latex])<\/li>\r\n<\/ol>\r\nThe constant value ([latex]a[\/latex], [latex]b[\/latex], or [latex]c[\/latex]) determines how far the plane is from the corresponding coordinate plane.\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The coordinate that equals a constant tells you which coordinate plane your plane is parallel to. If [latex]z = 5[\/latex], the plane is parallel to the [latex]xy[\/latex]-plane and located [latex]5[\/latex] units above it.<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]242489[\/ohm_question]<\/section>\r\n<h3>Spheres<\/h3>\r\nJust as a circle consists of all points in a plane equidistant from a center point, a <strong>sphere<\/strong> consists of all points in 3D space equidistant from a center point. The distance from the center to any point on the sphere is the <strong>radius<\/strong>.\r\n\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<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>equation of a sphere<\/h3>\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<\/section><section class=\"textbox recall\" aria-label=\"Recall\">Compare the 2D circle equation [latex](x-a)^2 + (y-b)^2 = r^2[\/latex] to the 3D sphere equation. The sphere equation simply adds the [latex](z-c)^2[\/latex] term for the third dimension.<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[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<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]328432[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Describe the set of points that satisfies [latex](x-4)(z-2)=0[\/latex], and graph the set.[reveal-answer q=\"887239643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"887239643\"]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).[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]<\/section><section class=\"textbox example\" aria-label=\"Example\">Describe the set of points in three-dimensional space that satisfies [latex](x+2)^{2}+(y-1)^{2}=4[\/latex], and graph the set.[reveal-answer q=\"767152709\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767152709\"]The [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).[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]<\/section>","rendered":"<h2 data-type=\"title\">Writing Equations in [latex]\\mathbb{R}^3[\/latex]<\/h2>\n<p class=\"whitespace-normal break-words\">Now that you can represent points in space and calculate distances between them, you can learn to write equations for geometric objects like lines, planes, and curved surfaces in three-dimensional space.<\/p>\n<p class=\"whitespace-normal break-words\">The same equation can represent completely different geometric objects depending on the dimensional space you&#8217;re working in. Consider the simple equation [latex]x = 0[\/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>In [latex]\\mathbb{R}[\/latex] (1D)<\/strong>: Describes a single point<\/li>\n<li class=\"whitespace-normal break-words\"><strong>In [latex]\\mathbb{R}^2[\/latex] (2D)<\/strong>: Describes a line (the [latex]y[\/latex]-axis)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>In [latex]\\mathbb{R}^3[\/latex] (3D)<\/strong>: Describes a plane (the [latex]yz[\/latex]-plane)<\/li>\n<\/ul>\n<figure id=\"attachment_5039\" aria-describedby=\"caption-attachment-5039\" style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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.<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">In 3D space, each coordinate plane has a simple equation:<\/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]yz[\/latex]-plane<\/strong>: [latex]x = 0[\/latex] (contains all points [latex](0, y, z)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>[latex]xy[\/latex]-plane<\/strong>: [latex]z = 0[\/latex] (contains all points [latex](x, y, 0)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>[latex]xz[\/latex]-plane<\/strong>: [latex]y = 0[\/latex] (contains all points [latex](x, 0, z)[\/latex])<\/li>\n<\/ul>\n<figure id=\"attachment_5044\" aria-describedby=\"caption-attachment-5044\" style=\"width: 732px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">Once you understand coordinate plane equations, you can write equations for any plane parallel to a coordinate plane. The key insight: when a plane is parallel to a coordinate plane, one coordinate remains constant for all points in that plane.<\/p>\n<p class=\"whitespace-normal break-words\">For example, if a plane is parallel to the [latex]xy[\/latex]-plane, every point in that plane has the same [latex]z[\/latex]-coordinate. Only the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates vary from point to point.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>equations of planes parallel to coordinate planes<\/h3>\n<ol id=\"fs-id1163724075425\" type=\"1\">\n<li class=\"whitespace-normal break-words\"><strong>Parallel to [latex]xy[\/latex]-plane<\/strong>: [latex]z = c[\/latex] (contains point [latex](a, b, c)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Parallel to [latex]xz[\/latex]-plane<\/strong>: [latex]y = b[\/latex] (contains point [latex](a, b, c)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Parallel to [latex]yz[\/latex]-plane<\/strong>: [latex]x = a[\/latex] (contains point [latex](a, b, c)[\/latex])<\/li>\n<\/ol>\n<p>The constant value ([latex]a[\/latex], [latex]b[\/latex], or [latex]c[\/latex]) determines how far the plane is from the corresponding coordinate plane.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The coordinate that equals a constant tells you which coordinate plane your plane is parallel to. If [latex]z = 5[\/latex], the plane is parallel to the [latex]xy[\/latex]-plane and located [latex]5[\/latex] units above it.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q485274093\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm242489\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=242489&theme=lumen&iframe_resize_id=ohm242489&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Spheres<\/h3>\n<p>Just as a circle consists of all points in a plane equidistant from a center point, a <strong>sphere<\/strong> consists of all points in 3D space equidistant from a center point. The distance from the center to any point on the sphere is the <strong>radius<\/strong>.<\/p>\n<figure id=\"attachment_5045\" aria-describedby=\"caption-attachment-5045\" style=\"width: 284px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\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<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>equation of a sphere<\/h3>\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<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">Compare the 2D circle equation [latex](x-a)^2 + (y-b)^2 = r^2[\/latex] to the 3D sphere equation. The sphere equation simply adds the [latex](z-c)^2[\/latex] term for the third dimension.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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<figure id=\"attachment_5046\" aria-describedby=\"caption-attachment-5046\" style=\"width: 473px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q942317736\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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<figure id=\"attachment_5047\" aria-describedby=\"caption-attachment-5047\" style=\"width: 550px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption id=\"caption-attachment-5047\" class=\"wp-caption-text\">Figure 5. Line segment [latex]PQ[\/latex].<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q177538250\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm328432\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=328432&theme=lumen&iframe_resize_id=ohm328432&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q887239643\">Show Solution<\/button><\/p>\n<div id=\"q887239643\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_5051\" aria-describedby=\"caption-attachment-5051\" style=\"width: 660px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/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)^{2}+(y-1)^{2}=4[\/latex], and graph the set.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q767152709\">Show Solution<\/button><\/p>\n<div id=\"q767152709\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_5053\" aria-describedby=\"caption-attachment-5053\" style=\"width: 501px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"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-875","chapter","type-chapter","status-publish","hentry"],"part":420,"module-header":"- Select Header -","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\/875","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":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/875\/revisions"}],"predecessor-version":[{"id":2904,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/875\/revisions\/2904"}],"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\/875\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=875"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=875"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=875"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=875"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}