{"id":5440,"date":"2022-06-02T18:17:19","date_gmt":"2022-06-02T18:17:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=5440"},"modified":"2022-10-21T00:00:17","modified_gmt":"2022-10-21T00:00:17","slug":"three-dimensional-coordinate-systems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/three-dimensional-coordinate-systems\/","title":{"raw":"Three-Dimensional Coordinate Systems","rendered":"Three-Dimensional Coordinate Systems"},"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\">Describe three-dimensional space mathematically.<\/span><\/li>\r\n \t<li><span class=\"os-abstract-content\">Locate points in space using coordinates.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1163723948509\">As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal [latex]x[\/latex]-axis and the vertical [latex]y[\/latex]-axis. We can add a third dimension, the [latex]z[\/latex]-axis, which is perpendicular to both the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.<\/p>\r\n\r\n<div id=\"fs-id1163723873271\" class=\"ui-has-child-title\" data-type=\"note\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nThe\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term64\" data-type=\"term\">three-dimensional rectangular coordinate system<\/span><\/strong>\u00a0consists of three perpendicular axes: the [latex]x[\/latex]-axis, the [latex]y[\/latex]-axis, and the [latex]z[\/latex]-axis. Because each axis is a number line representing all real numbers in [latex]\\mathbb{R}[\/latex] the three-dimensional system is often denoted by [latex]\\mathbb{R}^3[\/latex].\r\n\r\n<\/div>\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? The system displayed follows the<strong>\u00a0<span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term65\" data-type=\"term\">right-hand rule<\/span>.<\/strong> If we take our right hand and align the fingers with the positive [latex]x[\/latex]-axis, then curl the fingers so they point in the direction of the positive [latex]y[\/latex]-axis, our thumb points in the direction of the positive [latex]z[\/latex]-axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.\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<p id=\"fs-id1163723881242\">In two dimensions, we describe a point in the plane with the coordinates [latex](x, y)[\/latex]. Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, [latex]z[\/latex], is appended to indicate alignment with the [latex]z[\/latex]-axis: [latex](x, y, z)[\/latex]. A point in space is identified by all three coordinates (Figure 2). To plot the point [latex](x, y, z)[\/latex], go [latex]x[\/latex]\u00a0units along the [latex]x[\/latex]-axis, then [latex]y[\/latex]\u00a0units in the direction of the [latex]y[\/latex]-axis, then [latex]z[\/latex] units in the direction of the [latex]z[\/latex]-axis.<\/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]<\/div>\r\n<div><\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: locating points in space<\/h3>\r\nSketch the point [latex](1, -2, 3)[\/latex] in three-dimensional space.\r\n\r\n[reveal-answer q=\"893782736\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"893782736\"]\r\n\r\nTo 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).\r\n\r\n[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]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSketch the point [latex](-2, 3, -1)[\/latex] in three-dimensional space.\r\n\r\n[reveal-answer q=\"173554276\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173554276\"]\r\n\r\n[caption id=\"attachment_3480\" align=\"aligncenter\" width=\"376\"]<img class=\"wp-image-3480 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192236\/2-2-tryitans1.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn. One corner is labeled (-2, 3, -1).\" width=\"376\" height=\"359\" \/> Figure 4.\u00a0Sketch of point [latex](-2, 3, -1)[\/latex][\/caption][\/hidden-answer]<\/div>\r\n<p id=\"fs-id1163724039181\">In two-dimensional space, the coordinate plane is defined by a pair of perpendicular axes. These axes allow us to name any location within the plane. In three dimensions, we define\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term66\" data-type=\"term\">coordinate planes<\/span><\/strong>\u00a0by the coordinate axes, just as in two dimensions. There are three axes now, so there are three intersecting pairs of axes. Each pair of axes forms a coordinate plane: the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane (Figure 5). We define the [latex]xy[\/latex]-plane formally as the following set: [latex]\\{(x,y,0) :x,y\\in\\mathbb{R}\\}[\/latex]. Similarly, the [latex]xz[\/latex]-plane and the [latex]yz[\/latex]-plane are defined as [latex]\\{(x,0,z) :x,z\\in\\mathbb{R}\\}[\/latex] and [latex]\\{(0,y,z) :y,z\\in\\mathbb{R}\\}[\/latex], respectively.<\/p>\r\n<p id=\"fs-id1163723078206\">To visualize this, imagine you\u2019re building a house and are standing in a room with only two of the four walls finished. (Assume the two finished walls are adjacent to each other.) If you stand with your back to the corner where the two finished walls meet, facing out into the room, 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.<\/p>\r\n\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>In two dimensions, the coordinate axes partition the plane into four quadrants. Similarly, the coordinate planes divide space between them into eight regions about the origin, called\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term67\" data-type=\"term\">octants<\/span>.<\/strong> The octants fill [latex]\\mathbb{R}^3[\/latex] in the same way that quadrants fill [latex]\\mathbb{R}^2[\/latex], as shown in\u00a0Figure 6.<\/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>\r\n<p id=\"fs-id1163724037256\">Most work in three-dimensional space is a comfortable extension of the corresponding concepts in two dimensions. In this section, we use our knowledge of circles to describe spheres, then we expand our understanding of vectors to three dimensions. To accomplish these goals, we begin by adapting the distance formula to three-dimensional space.<\/p>\r\n<p id=\"fs-id1163724027502\">If two points lie in the same coordinate plane, then it is straightforward to calculate the distance between them. We that the distance [latex]d[\/latex] between two points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] in the [latex]xy[\/latex]-coordinate plane is given by the formula<\/p>\r\n\r\n<center>[latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} [\/latex].<\/center>\r\n<p id=\"fs-id1163723960539\">The formula for the distance between two points in space is a natural extension of this formula.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">theorem: the distance between two points in space<\/h3>\r\n\r\n<hr \/>\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><\/div>\r\nThe proof of this theorem is left as an exercise. (<em data-effect=\"italics\">Hint:<\/em>\u00a0First find the distance [latex]d_1[\/latex] between the points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_1)[\/latex] as shown in\u00a0Figure 7.)\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]\r\n<div class=\"textbox exercises\">\r\n<h3>Example: distance in space<\/h3>\r\nFind 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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nFind the distance between points [latex]P_1=(1, -5, 4)[\/latex] and\u00a0[latex]P_2=(4, -1, -1)[\/latex].\r\n\r\n[reveal-answer q=\"267361074\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"267361074\"]\r\n\r\n[latex]5\\sqrt{2}[\/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=7713625&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k_EYI3baAvI&amp;video_target=tpm-plugin-w5z1tf7u-k_EYI3baAvI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.12_transcript.html\">\u201cCP 2.12\u201d here (opens in new window)<\/a><\/center>Before moving on to the next section, let\u2019s get a feel for how [latex]\\mathbb{R}^3[\/latex] differs from\u00a0[latex]\\mathbb{R}^2[\/latex]. For example, in [latex]\\mathbb{R}^2[\/latex], lines that are not parallel must always intersect. This is not the case in\u00a0[latex]\\mathbb{R}^3[\/latex].For example, consider the line shown in\u00a0Figure 9<span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">. These two lines are not parallel, nor do they intersect.<\/span>\r\n\r\n&nbsp;\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\nYou can also have circles that are interconnected but have no points in common, as in\u00a0Figure 10.\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<p id=\"fs-id1163724036438\">We have a lot more flexibility working in three dimensions than we do if we stuck with only two dimensions.<\/p>\r\n\r\n<\/section><section id=\"fs-id1163723954119\" data-depth=\"1\"><\/section><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Describe three-dimensional space mathematically.<\/span><\/li>\n<li><span class=\"os-abstract-content\">Locate points in space using coordinates.<\/span><\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1163723948509\">As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal [latex]x[\/latex]-axis and the vertical [latex]y[\/latex]-axis. We can add a third dimension, the [latex]z[\/latex]-axis, which is perpendicular to both the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.<\/p>\n<div id=\"fs-id1163723873271\" class=\"ui-has-child-title\" data-type=\"note\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p>The\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term64\" data-type=\"term\">three-dimensional rectangular coordinate system<\/span><\/strong>\u00a0consists of three perpendicular axes: the [latex]x[\/latex]-axis, the [latex]y[\/latex]-axis, and the [latex]z[\/latex]-axis. Because each axis is a number line representing all real numbers in [latex]\\mathbb{R}[\/latex] the three-dimensional system is often denoted by [latex]\\mathbb{R}^3[\/latex].<\/p>\n<\/div>\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? The system displayed follows the<strong>\u00a0<span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term65\" data-type=\"term\">right-hand rule<\/span>.<\/strong> If we take our right hand and align the fingers with the positive [latex]x[\/latex]-axis, then curl the fingers so they point in the direction of the positive [latex]y[\/latex]-axis, our thumb points in the direction of the positive [latex]z[\/latex]-axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.<\/p>\n<div id=\"attachment_3483\" style=\"width: 714px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3483\" 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\" \/><\/p>\n<p 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.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163723881242\">In two dimensions, we describe a point in the plane with the coordinates [latex](x, y)[\/latex]. Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, [latex]z[\/latex], is appended to indicate alignment with the [latex]z[\/latex]-axis: [latex](x, y, z)[\/latex]. A point in space is identified by all three coordinates (Figure 2). To plot the point [latex](x, y, z)[\/latex], go [latex]x[\/latex]\u00a0units along the [latex]x[\/latex]-axis, then [latex]y[\/latex]\u00a0units in the direction of the [latex]y[\/latex]-axis, then [latex]z[\/latex] units in the direction of the [latex]z[\/latex]-axis.<\/p>\n<div id=\"CNX_Calc_Figure_12_02_002\" class=\"os-figure\">\n<div id=\"attachment_3484\" style=\"width: 281px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3484\" 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\" \/><\/p>\n<p 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.<\/p>\n<\/div>\n<\/div>\n<div><\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: locating points in space<\/h3>\n<p>Sketch the point [latex](1, -2, 3)[\/latex] in three-dimensional space.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q893782736\">Show Solution<\/span><\/p>\n<div id=\"q893782736\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<div id=\"attachment_3485\" style=\"width: 315px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3485\" 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\" \/><\/p>\n<p id=\"caption-attachment-3485\" class=\"wp-caption-text\">Figure 3. Sketching the point [latex](1,\u22122,3)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Sketch the point [latex](-2, 3, -1)[\/latex] in three-dimensional space.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q173554276\">Show Solution<\/span><\/p>\n<div id=\"q173554276\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"attachment_3480\" style=\"width: 386px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3480\" class=\"wp-image-3480 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28192236\/2-2-tryitans1.jpeg\" alt=\"This figure is the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn. One corner is labeled (-2, 3, -1).\" width=\"376\" height=\"359\" \/><\/p>\n<p id=\"caption-attachment-3480\" class=\"wp-caption-text\">Figure 4.\u00a0Sketch of point [latex](-2, 3, -1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163724039181\">In two-dimensional space, the coordinate plane is defined by a pair of perpendicular axes. These axes allow us to name any location within the plane. In three dimensions, we define\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term66\" data-type=\"term\">coordinate planes<\/span><\/strong>\u00a0by the coordinate axes, just as in two dimensions. There are three axes now, so there are three intersecting pairs of axes. Each pair of axes forms a coordinate plane: the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane (Figure 5). We define the [latex]xy[\/latex]-plane formally as the following set: [latex]\\{(x,y,0) :x,y\\in\\mathbb{R}\\}[\/latex]. Similarly, the [latex]xz[\/latex]-plane and the [latex]yz[\/latex]-plane are defined as [latex]\\{(x,0,z) :x,z\\in\\mathbb{R}\\}[\/latex] and [latex]\\{(0,y,z) :y,z\\in\\mathbb{R}\\}[\/latex], respectively.<\/p>\n<p id=\"fs-id1163723078206\">To visualize this, imagine you\u2019re building a house and are standing in a room with only two of the four walls finished. (Assume the two finished walls are adjacent to each other.) If you stand with your back to the corner where the two finished walls meet, facing out into the room, 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.<\/p>\n<div id=\"CNX_Calc_Figure_12_02_005\" class=\"os-figure\">\n<div id=\"attachment_3486\" style=\"width: 296px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3486\" 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\" \/><\/p>\n<p 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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div>In two dimensions, the coordinate axes partition the plane into four quadrants. Similarly, the coordinate planes divide space between them into eight regions about the origin, called\u00a0<strong><span id=\"1412f58c-b8e6-4517-ac82-71d0bdf7c1ba_term67\" data-type=\"term\">octants<\/span>.<\/strong> The octants fill [latex]\\mathbb{R}^3[\/latex] in the same way that quadrants fill [latex]\\mathbb{R}^2[\/latex], as shown in\u00a0Figure 6.<\/div>\n<div><\/div>\n<div>\n<div id=\"attachment_3487\" style=\"width: 419px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3487\" 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\" \/><\/p>\n<p id=\"caption-attachment-3487\" class=\"wp-caption-text\">Figure 6. Points that lie in octants have three nonzero coordinates.<\/p>\n<\/div>\n<\/div>\n<div>\n<p id=\"fs-id1163724037256\">Most work in three-dimensional space is a comfortable extension of the corresponding concepts in two dimensions. In this section, we use our knowledge of circles to describe spheres, then we expand our understanding of vectors to three dimensions. To accomplish these goals, we begin by adapting the distance formula to three-dimensional space.<\/p>\n<p id=\"fs-id1163724027502\">If two points lie in the same coordinate plane, then it is straightforward to calculate the distance between them. We that the distance [latex]d[\/latex] between two points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex] in the [latex]xy[\/latex]-coordinate plane 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}[\/latex].<\/div>\n<p id=\"fs-id1163723960539\">The formula for the distance between two points in space is a natural extension of this formula.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">theorem: the distance between two points in space<\/h3>\n<hr \/>\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<\/div>\n<p>The proof of this theorem is left as an exercise. (<em data-effect=\"italics\">Hint:<\/em>\u00a0First find the distance [latex]d_1[\/latex] between the points [latex](x_1, y_1, z_1)[\/latex] and [latex](x_2, y_2, z_1)[\/latex] as shown in\u00a0Figure 7.)<\/p>\n<div id=\"attachment_3488\" style=\"width: 491px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3488\" 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\" \/><\/p>\n<p 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.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: distance in space<\/h3>\n<p>Find the distance between points [latex]P_1=(3, -1, 5)[\/latex] and\u00a0[latex]P_2=(2, 1, -1)[\/latex].<\/p>\n<div id=\"attachment_3489\" style=\"width: 462px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3489\" 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\" \/><\/p>\n<p id=\"caption-attachment-3489\" class=\"wp-caption-text\">Figure 8. Find the distance between the two points.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q772679224\">Show Solution<\/span><\/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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Find the distance between points [latex]P_1=(1, -5, 4)[\/latex] and\u00a0[latex]P_2=(4, -1, -1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q267361074\">Show Solution<\/span><\/p>\n<div id=\"q267361074\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]5\\sqrt{2}[\/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=7713625&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k_EYI3baAvI&amp;video_target=tpm-plugin-w5z1tf7u-k_EYI3baAvI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.12_transcript.html\">\u201cCP 2.12\u201d here (opens in new window)<\/a><\/div>\n<p>Before moving on to the next section, let\u2019s get a feel for how [latex]\\mathbb{R}^3[\/latex] differs from\u00a0[latex]\\mathbb{R}^2[\/latex]. For example, in [latex]\\mathbb{R}^2[\/latex], lines that are not parallel must always intersect. This is not the case in\u00a0[latex]\\mathbb{R}^3[\/latex].For example, consider the line shown in\u00a0Figure 9<span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">. These two lines are not parallel, nor do they intersect.<\/span><\/p>\n<p>&nbsp;<\/p>\n<div id=\"attachment_4983\" style=\"width: 598px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4983\" 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\" \/><\/p>\n<p id=\"caption-attachment-4983\" class=\"wp-caption-text\">Figure 9. These two lines are not parallel, but still do not intersect.<\/p>\n<\/div>\n<p>You can also have circles that are interconnected but have no points in common, as in\u00a0Figure 10.<\/p>\n<div id=\"attachment_5030\" style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5030\" 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\" \/><\/p>\n<p id=\"caption-attachment-5030\" class=\"wp-caption-text\">Figure 10. These circles are interconnected, but have no points in common.<\/p>\n<\/div>\n<section id=\"fs-id1163723635408\" data-depth=\"1\">\n<p id=\"fs-id1163724036438\">We have a lot more flexibility working in three dimensions than we do if we stuck with only two dimensions.<\/p>\n<\/section>\n<section id=\"fs-id1163723954119\" data-depth=\"1\"><\/section>\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-5440\">\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 2.12. <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":8,"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 https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 2.12\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5440","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5440","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5440\/revisions"}],"predecessor-version":[{"id":6431,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5440\/revisions\/6431"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5440\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=5440"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=5440"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=5440"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=5440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}