{"id":500,"date":"2025-10-09T15:15:47","date_gmt":"2025-10-09T15:15:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=500"},"modified":"2026-08-25T15:31:17","modified_gmt":"2026-08-25T15:31:17","slug":"three-dimensional-space-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/three-dimensional-space-learn-it-1\/","title":{"raw":"Three-Dimensional Space: Learn It 1","rendered":"Three-Dimensional Space: Learn It 1"},"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<p class=\"whitespace-normal break-words\">Vectors work well for solving two-dimensional problems, but real life happens in three dimensions. Whether you're navigating mountainous terrain, analyzing the flight path of an aircraft, or studying molecular structures, you need tools that work in 3D space.<\/p>\r\nThe two-dimensional coordinate system uses two perpendicular axes: the horizontal [latex]x[\/latex]-axis and vertical [latex]y[\/latex]-axis. To work in three dimensions, we add a third axis\u2014the [latex]z[\/latex]-axis\u2014perpendicular to both existing axes.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>three-dimensional rectangular coordinate system<\/h3>\r\nConsists of three mutually perpendicular axes: the [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and [latex]z[\/latex]-axis. Since each axis represents all real numbers, we denote this system as [latex]\\mathbb{R}^3[\/latex].\r\n\r\n<\/section>\r\n<div id=\"fs-id1163723873271\" class=\"ui-has-child-title\" data-type=\"note\">\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">The Right-Hand Rule<\/h3>\r\nIn\u00a0Figure 1(a), the positive [latex]z[\/latex]-axis is shown above the plane containing the [latex]x[\/latex]- and [latex]y[\/latex]-axes. The positive [latex]x[\/latex]-axis appears to the left and the positive [latex]y[\/latex]-axis is to the right. A natural question to ask is: How was arrangement determined?\r\n<p class=\"whitespace-normal break-words\">The arrangement of the three axes follows a standard convention called the <strong>right-hand rule<\/strong>. Here's how it works:<\/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 your fingers toward the positive [latex]y[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Your thumb naturally points in the direction of the positive [latex]z[\/latex]-axis<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">This standard ensures consistency across mathematical applications. While left-hand rule systems exist, we always use the right-hand rule convention.<\/p>\r\n\r\n[caption id=\"attachment_3483\" align=\"aligncenter\" width=\"704\"]<img class=\"wp-image-3483 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192526\/2-2-1.jpeg\" alt=\"This figure has two images. The first is a 3-dimensional coordinate system. The x-axis is forward, the y-axis is horizontal to the left and right, and the z-axis is vertical. The second image is the 3-dimensional coordinate system axes with a right hand. The thumb is pointing towards positive z-axis, with the fingers wrapping around the z-axis from the positive x-axis to the positive y-axis.\" width=\"704\" height=\"386\" \/> Figure 1. (a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the [latex]z[\/latex]-axis, that is perpendicular to both the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.[\/caption]<\/div>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Plotting Points in 3D<\/h3>\r\n<p id=\"fs-id1163723881242\">In two dimensions, we describe a point in the plane with the coordinates [latex](x, y)[\/latex]. In three dimensions, you add a third coordinate, [latex]z[\/latex]: [latex](x, y, z)[\/latex]. A point in space is identified by all three coordinates (Figure 2).<\/p>\r\n<p class=\"whitespace-normal break-words\">To plot the point [latex](x, y, z)[\/latex]:<\/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\">Move [latex]x[\/latex] units along the [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move [latex]y[\/latex] units in the direction of the [latex]y[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Move [latex]z[\/latex] units in the direction of the [latex]z[\/latex]-axis<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">The intersection of these three movements locates your point in 3D space.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_12_02_002\" class=\"os-figure\">[caption id=\"attachment_3484\" align=\"aligncenter\" width=\"271\"]<img class=\"wp-image-3484 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192551\/2-2-2.jpeg\" alt=\"This figure is the positive octant of the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn with broken lines. One corner is labeled (x, y, z). The height of the box is labeled \u201cz units,\u201d the width is labeled \u201cx units\u201d and the length is labeled \u201cy units.\u201d\" width=\"271\" height=\"211\" \/> Figure 2. To plot the point [latex](x,y,z) [\/latex] go [latex] x [\/latex] units along the [latex]x[\/latex]-axis, then [latex]y[\/latex] units in the direction of the [latex]y[\/latex]-axis, then [latex] z[\/latex] units in the direction of the z-axis.[\/caption]<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0Three-dimensional plotting can be challenging to visualize on paper. Try using physical objects or online 3D graphing tools to better understand how points relate to the three axes<\/section><\/div>\r\n<div><section class=\"textbox example\" aria-label=\"Example\">Sketch the point [latex](1, -2, 3)[\/latex] in three-dimensional space.[reveal-answer q=\"893782736\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"893782736\"]To sketch a point, start by sketching three sides of a rectangular prism along the coordinate axes: one unit in the positive [latex]x[\/latex] direction, 2 units in the negative [latex]y[\/latex] direction, and 3 units in the positive [latex]z[\/latex] direction. Complete the prism to plot the point (Figure 3).[caption id=\"attachment_3485\" align=\"aligncenter\" width=\"305\"]<img class=\"wp-image-3485 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192628\/2-2-3.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. In the fourth octant there is a rectangular solid drawn. One corner is labeled (1, -2, 3).\" width=\"305\" height=\"359\" \/> Figure 3. Sketching the point [latex](1,\u22122,3)[\/latex].[\/caption][\/hidden-answer]<\/section>\r\n<div><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]328430[\/ohm_question]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Coordinate Planes<\/h3>\r\n<p class=\"whitespace-normal break-words\">Just as two perpendicular axes define a coordinate plane in 2D, pairs of axes define coordinate planes in 3D. With three axes, we get three coordinate planes:<\/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>: Contains the [latex]x[\/latex] and [latex]y[\/latex] axes, where [latex]z = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]xz[\/latex]-plane<\/strong>: Contains the [latex]x[\/latex] and [latex]z[\/latex] axes, where [latex]y = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>[latex]yz[\/latex]-plane<\/strong>: Contains the [latex]y[\/latex] and [latex]z[\/latex] axes, where [latex]x = 0[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Formally, these planes are defined as:<\/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]xy[\/latex]-plane: [latex]{(x,y,0) : x,y \\in \\mathbb{R}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]xz[\/latex]-plane: [latex]{(x,0,z) : x,z \\in \\mathbb{R}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]yz[\/latex]-plane: [latex]{(0,y,z) : y,z \\in \\mathbb{R}}[\/latex]<\/li>\r\n<\/ul>\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">To visualize this, imagine standing in a corner room with two finished walls meeting at a right angle. If you face outward from the corner: the floor is the [latex]xy[\/latex]-plane, the wall to your right is the [latex]xz[\/latex]-plane, and the wall to your left is the [latex]yz[\/latex]-plane.<\/section>\r\n<div id=\"CNX_Calc_Figure_12_02_005\" class=\"os-figure\">[caption id=\"attachment_3486\" align=\"aligncenter\" width=\"286\"]<img class=\"wp-image-3486 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192719\/2-2-4.jpeg\" alt=\"This figure is the first octant of a 3-dimensional coordinate system. Also, there are the x y-plane represented with a rectangle with the x and y axes on the plane. There is also the x z-plane on the x and z axes and the y z-plane on the y and z axes.\" width=\"286\" height=\"291\" \/> Figure 5. The plane containing the [latex]x[\/latex]- and [latex]y [\/latex]-axes is called the [latex]xy [\/latex]-plane. The plane containing the [latex]x [\/latex]- and [latex]z [\/latex]-axes is called the [latex]xz [\/latex]-plane, and the [latex]y [\/latex]- and [latex]z [\/latex]-axes define the [latex]yz [\/latex]-plane.[\/caption]<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n<p class=\"whitespace-normal break-words\">In 2D, the coordinate axes divide the plane into four quadrants. In 3D, the three coordinate planes divide space into eight regions called <strong>octants<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n[caption id=\"attachment_3487\" align=\"aligncenter\" width=\"409\"]<img class=\"wp-image-3487 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192748\/2-2-5.jpeg\" alt=\"This figure is the 3-dimensional coordinate system with the first octant labeled with a roman numeral I, I, II, III, IV, V, VI, VII, and VIII. Also, for each quadrant there are the signs of the values of x, y, and z. They are: I (+, +, +); 2nd (-, +, +); 3rd (-, -, +); 4th (+, -, +); 5th (+, +, -); 6th (-, +, -); 7th (-, -, -); and 8th (+, -, -).\" width=\"409\" height=\"415\" \/> Figure 6. Points that lie in octants have three nonzero coordinates.[\/caption]\r\n\r\n<\/div>\r\n<div><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]328429[\/ohm_question]<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Distance Formula in Three Dimensions<\/h3>\r\n<p class=\"whitespace-normal break-words\">The distance formula extends naturally from 2D to 3D. In two dimensions, the distance between points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] is:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[\/latex]<\/p>\r\n<p id=\"fs-id1163724027502\">The distance [latex]d[\/latex] between points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_2)[\/latex] is:<\/p>\r\n<p style=\"text-align: center;\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[\/latex]<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>distance formula in 3D<\/h3>\r\n<p id=\"fs-id1163723751036\">The distance [latex]d[\/latex] between points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_2)[\/latex] is given by the formula:<\/p>\r\n\r\n<center>[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} [\/latex].<\/center><\/section>This formula represents the length of the diagonal of a rectangular prism (box) with the two points as opposite corners.\r\n\r\n[caption id=\"attachment_3488\" align=\"aligncenter\" width=\"481\"]<img class=\"wp-image-3488 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192817\/2-2-6.jpeg\" alt=\"This figure is a rectangular prism. The lower, left back corner is labeled \u201cP sub 1=(x sub 1,y sub 1,z sub 1). The lower front right corner is labeled \u201c(x sub 2, y sub 2, z sub 1)\u201d. There is a line between P sub 1 and P sub 2 and is labeled \u201cd sub 1\u201d. The upper front right corner is labeled \u201cP sub 2=(x sub 2,y sub 2,z sub 2).\u201d There is a line from P sub 1 to P sub 2 and is labeled \u201cd (P sub 1,P sub 2).\u201d The front right vertical side is labeled \u201c|z sub 2-z sub 1|\u201d.\" width=\"481\" height=\"224\" \/> Figure 7. The distance between [latex]P_1[\/latex] and [latex]P_2[\/latex] is the length of the diagonal of the rectangular prism having [latex]P_1[\/latex] and [latex]P_2[\/latex] as opposite corners.[\/caption]<section class=\"textbox example\" aria-label=\"Example\">Find the distance between points [latex]P_1=(3, -1, 5)[\/latex] and\u00a0[latex]P_2=(2, 1, -1)[\/latex].\r\n\r\n[caption id=\"attachment_3489\" align=\"aligncenter\" width=\"452\"]<img class=\"wp-image-3489 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192850\/2-2-7.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. There are two points. The first is labeled \u201cP sub 1(3, -1, 5)\u201d and the second is labeled \u201cP sub 2(2, 1, -1)\u201d. There is a line segment between the two points.\" width=\"452\" height=\"470\" \/> Figure 8. Find the distance between the two points.[\/caption]\r\n\r\n[reveal-answer q=\"772679224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"772679224\"]\r\n\r\nSubstitute values directly into the distance formula:\r\n\r\n<center>\r\n[latex]\r\n\\begin{align*}\r\nd(P1,P2) &amp;= \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\\\\\r\n&amp;= \\sqrt{(2 - 3)^2 + (1 - (-1))^2 + (-1 - 5)^2} \\\\\r\n&amp;= \\sqrt{(-1)^2 + 2^2 + (-6)^2} \\\\\r\n&amp;= \\sqrt{41} \\\\\r\n\\end{align*}\r\n[\/latex]<\/center>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]328431[\/ohm_question]\r\n\r\n<\/section>\r\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Key Differences in Three Dimensions<\/h3>\r\nBefore moving on to the next section, let\u2019s get a feel for how [latex]\\mathbb{R}^3[\/latex] differs from [latex]\\mathbb{R}^2[\/latex].\r\n<p class=\"whitespace-normal break-words\">Three-dimensional space offers more geometric flexibility than 2D:<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>In 2D<\/strong>: Non-parallel lines must intersect.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>In 3D<\/strong>: Non-parallel lines can be <strong>skew lines<\/strong>\u2014they don't intersect and aren't parallel.<\/p>\r\n\r\n\r\n[caption id=\"attachment_4983\" align=\"aligncenter\" width=\"588\"]<img class=\"size-full wp-image-4983\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26170610\/4.20.png\" alt=\"This figure is the 3-dimensional coordinate system. There is a line drawn at z = 3. It is parallel to the x y-plane. There is also a line drawn at y = 2. It is parallel to the x-axis.\" width=\"588\" height=\"507\" \/> Figure 9. These two lines are not parallel, but still do not intersect.[\/caption]\r\n\r\n&nbsp;\r\n\r\n<strong>In 2D<\/strong>: Circles either intersect, are tangent, or are completely separate.\r\n\r\n<strong>In 3D<\/strong>: Circles can be interconnected (like links in a chain) without sharing any points.\r\n\r\n[caption id=\"attachment_5030\" align=\"aligncenter\" width=\"488\"]<img class=\"size-full wp-image-5030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26185750\/Figure-2.31.jpg\" alt=\"This figure is the 3-dimensional coordinate system. There are two circles drawn. The first circle is centered around the z-axis, at z = 1. The second circle has the positive x-axis as its diameter. It intersects the x-axis at x = 0 and x = 6. It is vertical.\" width=\"488\" height=\"511\" \/> Figure 10. These circles are interconnected, but have no points in common.[\/caption]\r\n\r\n<section id=\"fs-id1163723635408\" data-depth=\"1\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\r\n<p class=\"whitespace-normal break-words\">This increased flexibility makes 3D geometry richer and more complex than 2D geometry.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>","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<p class=\"whitespace-normal break-words\">Vectors work well for solving two-dimensional problems, but real life happens in three dimensions. Whether you&#8217;re navigating mountainous terrain, analyzing the flight path of an aircraft, or studying molecular structures, you need tools that work in 3D space.<\/p>\n<p>The two-dimensional coordinate system uses two perpendicular axes: the horizontal [latex]x[\/latex]-axis and vertical [latex]y[\/latex]-axis. To work in three dimensions, we add a third axis\u2014the [latex]z[\/latex]-axis\u2014perpendicular to both existing axes.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>three-dimensional rectangular coordinate system<\/h3>\n<p>Consists of three mutually perpendicular axes: the [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and [latex]z[\/latex]-axis. Since each axis represents all real numbers, we denote this system as [latex]\\mathbb{R}^3[\/latex].<\/p>\n<\/section>\n<div id=\"fs-id1163723873271\" class=\"ui-has-child-title\" data-type=\"note\">\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">The Right-Hand Rule<\/h3>\n<p>In\u00a0Figure 1(a), the positive [latex]z[\/latex]-axis is shown above the plane containing the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-axes. The positive [latex]x[\/latex]-axis appears to the left and the positive [latex]y[\/latex]-axis is to the right. A natural question to ask is: How was arrangement determined?<\/p>\n<p class=\"whitespace-normal break-words\">The arrangement of the three axes follows a standard convention called the <strong>right-hand rule<\/strong>. Here&#8217;s how it works:<\/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 your fingers toward the positive [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Your thumb naturally points in the direction of the positive [latex]z[\/latex]-axis<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">This standard ensures consistency across mathematical applications. While left-hand rule systems exist, we always use the right-hand rule convention.<\/p>\n<figure id=\"attachment_3483\" aria-describedby=\"caption-attachment-3483\" style=\"width: 704px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3483 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192526\/2-2-1.jpeg\" alt=\"This figure has two images. The first is a 3-dimensional coordinate system. The x-axis is forward, the y-axis is horizontal to the left and right, and the z-axis is vertical. The second image is the 3-dimensional coordinate system axes with a right hand. The thumb is pointing towards positive z-axis, with the fingers wrapping around the z-axis from the positive x-axis to the positive y-axis.\" width=\"704\" height=\"386\" \/><figcaption id=\"caption-attachment-3483\" class=\"wp-caption-text\">Figure 1. (a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the [latex]z[\/latex]-axis, that is perpendicular to both the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.<\/figcaption><\/figure>\n<\/div>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Plotting Points in 3D<\/h3>\n<p id=\"fs-id1163723881242\">In two dimensions, we describe a point in the plane with the coordinates [latex](x, y)[\/latex]. In three dimensions, you add a third coordinate, [latex]z[\/latex]: [latex](x, y, z)[\/latex]. A point in space is identified by all three coordinates (Figure 2).<\/p>\n<p class=\"whitespace-normal break-words\">To plot the point [latex](x, y, z)[\/latex]:<\/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\">Move [latex]x[\/latex] units along the [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Move [latex]y[\/latex] units in the direction of the [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Move [latex]z[\/latex] units in the direction of the [latex]z[\/latex]-axis<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">The intersection of these three movements locates your point in 3D space.<\/p>\n<div id=\"CNX_Calc_Figure_12_02_002\" class=\"os-figure\">\n<figure id=\"attachment_3484\" aria-describedby=\"caption-attachment-3484\" style=\"width: 271px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3484 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192551\/2-2-2.jpeg\" alt=\"This figure is the positive octant of the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn with broken lines. One corner is labeled (x, y, z). The height of the box is labeled \u201cz units,\u201d the width is labeled \u201cx units\u201d and the length is labeled \u201cy units.\u201d\" width=\"271\" height=\"211\" \/><figcaption id=\"caption-attachment-3484\" class=\"wp-caption-text\">Figure 2. To plot the point [latex](x,y,z) [\/latex] go [latex] x [\/latex] units along the [latex]x[\/latex]-axis, then [latex]y[\/latex] units in the direction of the [latex]y[\/latex]-axis, then [latex] z[\/latex] units in the direction of the z-axis.<\/figcaption><\/figure>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0Three-dimensional plotting can be challenging to visualize on paper. Try using physical objects or online 3D graphing tools to better understand how points relate to the three axes<\/section>\n<\/div>\n<div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the point [latex](1, -2, 3)[\/latex] in three-dimensional space.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q893782736\">Show Solution<\/button><\/p>\n<div id=\"q893782736\" class=\"hidden-answer\" style=\"display: none\">To sketch a point, start by sketching three sides of a rectangular prism along the coordinate axes: one unit in the positive [latex]x[\/latex] direction, 2 units in the negative [latex]y[\/latex] direction, and 3 units in the positive [latex]z[\/latex] direction. Complete the prism to plot the point (Figure 3).<\/p>\n<figure id=\"attachment_3485\" aria-describedby=\"caption-attachment-3485\" style=\"width: 305px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3485 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192628\/2-2-3.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. In the fourth octant there is a rectangular solid drawn. One corner is labeled (1, -2, 3).\" width=\"305\" height=\"359\" \/><figcaption id=\"caption-attachment-3485\" class=\"wp-caption-text\">Figure 3. Sketching the point [latex](1,\u22122,3)[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm328430\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=328430&theme=lumen&iframe_resize_id=ohm328430&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Coordinate Planes<\/h3>\n<p class=\"whitespace-normal break-words\">Just as two perpendicular axes define a coordinate plane in 2D, pairs of axes define coordinate planes in 3D. With three axes, we get three coordinate planes:<\/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>: Contains the [latex]x[\/latex] and [latex]y[\/latex] axes, where [latex]z = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>[latex]xz[\/latex]-plane<\/strong>: Contains the [latex]x[\/latex] and [latex]z[\/latex] axes, where [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>[latex]yz[\/latex]-plane<\/strong>: Contains the [latex]y[\/latex] and [latex]z[\/latex] axes, where [latex]x = 0[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Formally, these planes are defined as:<\/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]xy[\/latex]-plane: [latex]{(x,y,0) : x,y \\in \\mathbb{R}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]xz[\/latex]-plane: [latex]{(x,0,z) : x,z \\in \\mathbb{R}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]yz[\/latex]-plane: [latex]{(0,y,z) : y,z \\in \\mathbb{R}}[\/latex]<\/li>\n<\/ul>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">To visualize this, imagine standing in a corner room with two finished walls meeting at a right angle. If you face outward from the corner: the floor is the [latex]xy[\/latex]-plane, the wall to your right is the [latex]xz[\/latex]-plane, and the wall to your left is the [latex]yz[\/latex]-plane.<\/section>\n<div id=\"CNX_Calc_Figure_12_02_005\" class=\"os-figure\">\n<figure id=\"attachment_3486\" aria-describedby=\"caption-attachment-3486\" style=\"width: 286px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3486 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192719\/2-2-4.jpeg\" alt=\"This figure is the first octant of a 3-dimensional coordinate system. Also, there are the x y-plane represented with a rectangle with the x and y axes on the plane. There is also the x z-plane on the x and z axes and the y z-plane on the y and z axes.\" width=\"286\" height=\"291\" \/><figcaption id=\"caption-attachment-3486\" class=\"wp-caption-text\">Figure 5. The plane containing the [latex]x[\/latex]&#8211; and [latex]y [\/latex]-axes is called the [latex]xy [\/latex]-plane. The plane containing the [latex]x [\/latex]&#8211; and [latex]z [\/latex]-axes is called the [latex]xz [\/latex]-plane, and the [latex]y [\/latex]&#8211; and [latex]z [\/latex]-axes define the [latex]yz [\/latex]-plane.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div><\/div>\n<div>\n<p class=\"whitespace-normal break-words\">In 2D, the coordinate axes divide the plane into four quadrants. In 3D, the three coordinate planes divide space into eight regions called <strong>octants<\/strong>.<\/p>\n<\/div>\n<div><\/div>\n<div>\n<figure id=\"attachment_3487\" aria-describedby=\"caption-attachment-3487\" style=\"width: 409px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3487 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192748\/2-2-5.jpeg\" alt=\"This figure is the 3-dimensional coordinate system with the first octant labeled with a roman numeral I, I, II, III, IV, V, VI, VII, and VIII. Also, for each quadrant there are the signs of the values of x, y, and z. They are: I (+, +, +); 2nd (-, +, +); 3rd (-, -, +); 4th (+, -, +); 5th (+, +, -); 6th (-, +, -); 7th (-, -, -); and 8th (+, -, -).\" width=\"409\" height=\"415\" \/><figcaption id=\"caption-attachment-3487\" class=\"wp-caption-text\">Figure 6. Points that lie in octants have three nonzero coordinates.<\/figcaption><\/figure>\n<\/div>\n<div>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm328429\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=328429&theme=lumen&iframe_resize_id=ohm328429&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Distance Formula in Three Dimensions<\/h3>\n<p class=\"whitespace-normal break-words\">The distance formula extends naturally from 2D to 3D. In two dimensions, the distance between points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] is:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[\/latex]<\/p>\n<p id=\"fs-id1163724027502\">The distance [latex]d[\/latex] between points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_2)[\/latex] is:<\/p>\n<p style=\"text-align: center;\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>distance formula in 3D<\/h3>\n<p id=\"fs-id1163723751036\">The distance [latex]d[\/latex] between points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_2)[\/latex] is given by the formula:<\/p>\n<div style=\"text-align: center;\">[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[\/latex].<\/div>\n<\/section>\n<p>This formula represents the length of the diagonal of a rectangular prism (box) with the two points as opposite corners.<\/p>\n<figure id=\"attachment_3488\" aria-describedby=\"caption-attachment-3488\" style=\"width: 481px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3488 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192817\/2-2-6.jpeg\" alt=\"This figure is a rectangular prism. The lower, left back corner is labeled \u201cP sub 1=(x sub 1,y sub 1,z sub 1). The lower front right corner is labeled \u201c(x sub 2, y sub 2, z sub 1)\u201d. There is a line between P sub 1 and P sub 2 and is labeled \u201cd sub 1\u201d. The upper front right corner is labeled \u201cP sub 2=(x sub 2,y sub 2,z sub 2).\u201d There is a line from P sub 1 to P sub 2 and is labeled \u201cd (P sub 1,P sub 2).\u201d The front right vertical side is labeled \u201c|z sub 2-z sub 1|\u201d.\" width=\"481\" height=\"224\" \/><figcaption id=\"caption-attachment-3488\" class=\"wp-caption-text\">Figure 7. The distance between [latex]P_1[\/latex] and [latex]P_2[\/latex] is the length of the diagonal of the rectangular prism having [latex]P_1[\/latex] and [latex]P_2[\/latex] as opposite corners.<\/figcaption><\/figure>\n<section class=\"textbox example\" aria-label=\"Example\">Find the distance between points [latex]P_1=(3, -1, 5)[\/latex] and\u00a0[latex]P_2=(2, 1, -1)[\/latex].<\/p>\n<figure id=\"attachment_3489\" aria-describedby=\"caption-attachment-3489\" style=\"width: 452px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3489 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192850\/2-2-7.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. There are two points. The first is labeled \u201cP sub 1(3, -1, 5)\u201d and the second is labeled \u201cP sub 2(2, 1, -1)\u201d. There is a line segment between the two points.\" width=\"452\" height=\"470\" \/><figcaption id=\"caption-attachment-3489\" class=\"wp-caption-text\">Figure 8. Find the distance between the two points.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q772679224\">Show Solution<\/button><\/p>\n<div id=\"q772679224\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute values directly into the distance formula:<\/p>\n<div style=\"text-align: center;\">\n[latex]\\begin{align*}  d(P1,P2) &= \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\\\\  &= \\sqrt{(2 - 3)^2 + (1 - (-1))^2 + (-1 - 5)^2} \\\\  &= \\sqrt{(-1)^2 + 2^2 + (-6)^2} \\\\  &= \\sqrt{41} \\\\  \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm328431\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=328431&theme=lumen&iframe_resize_id=ohm328431&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Key Differences in Three Dimensions<\/h3>\n<p>Before moving on to the next section, let\u2019s get a feel for how [latex]\\mathbb{R}^3[\/latex] differs from [latex]\\mathbb{R}^2[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Three-dimensional space offers more geometric flexibility than 2D:<\/p>\n<p class=\"whitespace-normal break-words\"><strong>In 2D<\/strong>: Non-parallel lines must intersect.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>In 3D<\/strong>: Non-parallel lines can be <strong>skew lines<\/strong>\u2014they don&#8217;t intersect and aren&#8217;t parallel.<\/p>\n<figure id=\"attachment_4983\" aria-describedby=\"caption-attachment-4983\" style=\"width: 588px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-4983\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26170610\/4.20.png\" alt=\"This figure is the 3-dimensional coordinate system. There is a line drawn at z = 3. It is parallel to the x y-plane. There is also a line drawn at y = 2. It is parallel to the x-axis.\" width=\"588\" height=\"507\" \/><figcaption id=\"caption-attachment-4983\" class=\"wp-caption-text\">Figure 9. These two lines are not parallel, but still do not intersect.<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p><strong>In 2D<\/strong>: Circles either intersect, are tangent, or are completely separate.<\/p>\n<p><strong>In 3D<\/strong>: Circles can be interconnected (like links in a chain) without sharing any points.<\/p>\n<figure id=\"attachment_5030\" aria-describedby=\"caption-attachment-5030\" style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/26185750\/Figure-2.31.jpg\" alt=\"This figure is the 3-dimensional coordinate system. There are two circles drawn. The first circle is centered around the z-axis, at z = 1. The second circle has the positive x-axis as its diameter. It intersects the x-axis at x = 0 and x = 6. It is vertical.\" width=\"488\" height=\"511\" \/><figcaption id=\"caption-attachment-5030\" class=\"wp-caption-text\">Figure 10. These circles are interconnected, but have no points in common.<\/figcaption><\/figure>\n<section id=\"fs-id1163723635408\" data-depth=\"1\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 !gap-3.5\">\n<p class=\"whitespace-normal break-words\">This increased flexibility makes 3D geometry richer and more complex than 2D geometry.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":15,"menu_order":11,"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-500","chapter","type-chapter","status-publish","hentry"],"part":420,"module-header":"learn_it","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\/500","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":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/500\/revisions"}],"predecessor-version":[{"id":2903,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/500\/revisions\/2903"}],"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\/500\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=500"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=500"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=500"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}