{"id":27599,"date":"2021-08-26T13:00:18","date_gmt":"2021-08-26T13:00:18","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27599"},"modified":"2024-08-07T08:28:57","modified_gmt":"2024-08-07T08:28:57","slug":"solved-problems-of-plane-mirrors","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/solved-problems-of-plane-mirrors\/","title":{"rendered":"Solved Problems of Plane Mirrors"},"content":{"rendered":"<p><center><\/p>\n<h1>Plane Mirrors, Solved Problems<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nIn this class, we will review some solved problems of plane mirrors. The angle of reflection <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> is determined based on the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> between two plane mirrors joined by a hinge, and specific examples are calculated. Critical values of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> are examined so that the ray bounces once on each mirror, and the formula for <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> is validated. Additionally, angles of incidence are identified that cause the ray to return on itself, calculating a sequence of return angles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_n = n\\theta<\/span><\/span>.<br \/>\n<\/em><br \/>\n<u><strong>Learning Objectives<\/strong><\/u><br \/>\nBy the end of this class, the student will be able to:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the fundamental formulas of plane mirror optics.<\/li>\n<li><strong>Apply<\/strong> the law of reflection in problems involving plane mirrors.<\/li>\n<li><strong>Determine<\/strong> the angle of reflection <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> based on the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> between two plane mirrors.<\/li>\n<li><strong>Analyze<\/strong> the limits of the formulas for mirrors and their validity conditions.<\/li>\n<\/ol>\n<p><strong>TABLE OF CONTENTS<\/strong><br \/>\n<a href=\"#1\"><strong>Introduction<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Mirrors Joined by a Hinge<\/strong><\/a><br \/>\n<a href=\"#3\">Examining the Limits of Reasoning<\/a><br \/>\n<a href=\"#4\">Return Angles<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/sX--0tertWI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p style=\"text-align: justify; color: #000000;\">In the <a href=\"http:\/\/toposuranos.com\/material\/en\/reflection-in-plane-and-spherical-mirrors\/\" rel=\"noopener\" target=\"_blank\">previous class<\/a>, we reviewed most of the formulas related to the optics of plane and spherical mirrors; however, to achieve a better understanding of these topics, it is necessary to review how these appear in solving problems associated with these topics. Therefore, we will dedicate this part exclusively to reviewing the solution of some problems. This time we will focus exclusively on plane mirrors.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Mirrors Joined by a Hinge<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=sX--0tertWI&amp;t=64s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Two plane mirrors joined by<\/span><\/strong><\/a> one end hold an angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta.<\/span><\/span> If a light ray hits one of the mirrors at an angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> relative to the normal so that the light bounces only once on each mirror and intersects itself forming an angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma:<\/span><\/span><\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-CA7kLZjaxyo\/YSMSe1ea97I\/AAAAAAAAFc8\/i5snILhLzT8XlVM88Hs8JDZiJvum5zmlgCLcBGAsYHQ\/s0\/ESPEJOPLANO1.PNG\" width=\"400\" height=\"100\" alt=\"angles in plane mirrors\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-CA7kLZjaxyo\/YSMSe1ea97I\/AAAAAAAAFc8\/i5snILhLzT8XlVM88Hs8JDZiJvum5zmlgCLcBGAsYHQ\/s0\/ESPEJOPLANO1.PNG\" width=\"400\" height=\"100\" alt=\"angles in plane mirrors\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li type=\"a\">Find a formula to determine the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> in terms of the other data.<\/li>\n<li type=\"a\">If the light ray hits the first mirror at an angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=30^o<\/span><\/span> and the angle between the mirrors is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta=50^o<\/span><\/span>, what will the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> be?<\/li>\n<\/ol>\n<span class=\"collapseomatic \" id=\"id6abf26814e9bf\"  tabindex=\"0\" title=\"SOLUTION\"    >SOLUTION<\/span><div id=\"target-id6abf26814e9bf\" class=\"collapseomatic_content \">\n<ol style=\"text-align: justify; color: #000000;\">\n<li type=\"a\">Defining the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> between the normal of the second mirror and the light ray reflected from the first mirror, and using the law of reflection in plane mirrors, we can complete the figure as follows:<br \/>\n<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-zheSpWUbupU\/YSMU24o_WkI\/AAAAAAAAFdE\/iOtRNY_vBWMWaY24ycU8llcenhsBRVpDQCLcBGAsYHQ\/s0\/espejoplano2.PNG\" width=\"400\" height=\"100\" alt=\"angle between plane mirrors\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-zheSpWUbupU\/YSMU24o_WkI\/AAAAAAAAFdE\/iOtRNY_vBWMWaY24ycU8llcenhsBRVpDQCLcBGAsYHQ\/s0\/espejoplano2.PNG\" width=\"400\" height=\"100\" alt=\"angle between plane mirrors\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\nWith this in mind, it is now possible to perform the following reasoning:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 50px;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"width: 350px;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(90^o - \\alpha) + (90^o - \\beta) + \\theta = 180^o<\/span><\/span><\/td>\n<td>; Because the sum of the interior angles of a triangle is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">180^o<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\alpha + \\beta = \\theta <\/span><\/span><\/td>\n<td> <\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50px;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2\\alpha +2\\beta + \\gamma = 180 <\/span><\/span><\/td>\n<td>; Because the sum of the interior angles of a triangle is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">180^o<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv<\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\gamma = 180 - 2(\\alpha + \\beta)<\/span><\/span><\/td>\n<td> <\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50px;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{\\gamma = 180 - 2\\theta}<\/span><\/span><\/td>\n<td>; From (1,2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Therefore, it is inferred that the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/span> will only be a function of the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> formed by the mirrors, and its formula will be <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma(\\theta) = 180^0 - 2\\theta<\/span><\/span><\/li>\n<li type=\"a\">Based on the reasoning in the previous part, we have that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma = 180^o - 2\\cdot 50^o = 80^o <\/span><\/span><\/li>\n<\/ol>\n<\/div>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Examining the Limits of Reasoning<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=sX--0tertWI&amp;t=464s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The previous exercise has a delicate<\/span><\/strong><\/a> problem. If you observe the statement, you will see that it requires that the light ray must only bounce once on each mirror; however, not any value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> will serve for that to happen. Find the values of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> that satisfy such a condition and thus allow the formula obtained in the previous exercise to be valid.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-FIE6EKNxu1Q\/YSM34HUa-NI\/AAAAAAAAFdM\/DVM38fUBrxggrGsXoJcAI4SgDC2u5gm1gCLcBGAsYHQ\/s0\/espejosplanos3.PNG\" width=\"1021\" height=\"485\" alt=\"Light rays bouncing on plane mirrors\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-FIE6EKNxu1Q\/YSM34HUa-NI\/AAAAAAAAFdM\/DVM38fUBrxggrGsXoJcAI4SgDC2u5gm1gCLcBGAsYHQ\/s0\/espejosplanos3.PNG\" width=\"1021\" height=\"485\" alt=\"Light rays bouncing on plane mirrors\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><span class=\"collapseomatic \" id=\"id6abf26814eafe\"  tabindex=\"0\" title=\"SOLUTION\"    >SOLUTION<\/span><div id=\"target-id6abf26814eafe\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\">We have that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> reaches the \u00abcritical\u00bb value when it makes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta=0^o;<\/span><\/span> and when this happens, we can take an angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> that allows the following reasoning:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-b_rcYFH-moo\/YSM--pA_6sI\/AAAAAAAAFdU\/VQRBmlh7I-oAhjuWi83GOphA1JKQHrrbACLcBGAsYHQ\/s0\/espejoplano4.PNG\" width=\"288\" height=\"319\" alt=\"ray bouncing on plane mirrors with critical angle\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-b_rcYFH-moo\/YSM--pA_6sI\/AAAAAAAAFdU\/VQRBmlh7I-oAhjuWi83GOphA1JKQHrrbACLcBGAsYHQ\/s0\/espejoplano4.PNG\" width=\"288\" height=\"319\" alt=\"ray bouncing on plane mirrors with critical angle\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">The following two equations must occur:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha + x = 90^o<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta + x = 90^o<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And this is only possible if:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = \\theta<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">That is to say: the value <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=\\theta<\/span><\/span> is the critical incidence angle such that if it is exceeded, then the ray will bounce more than twice on a mirror and, consequently, invalidate the formula obtained in the previous exercise. Based on these results, we can correct the result of the previous exercise by writing:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma(\\theta, \\alpha) = 180^0 - 2\\theta \\;\\;\\;\\; ; \\;\\;\\;\\; \\alpha \\in ]0,\\theta[ <\/span><\/span><\/p>\n<\/div>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Return Angles<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=sX--0tertWI&amp;t=809s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">From these results, we can see that,<\/span><\/strong><\/a> for certain incidence angles, the light ray returns on itself. This happens when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0^o<\/span><\/span> or when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = \\theta,<\/span><\/span> where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span><\/span> is the angle formed between the two plane mirrors. Are there more return angles?; and if there are, how can they be calculated?<\/p>\n<span class=\"collapseomatic \" id=\"id6abf26814ebd1\"  tabindex=\"0\" title=\"SOLUTION\"    >SOLUTION<\/span><div id=\"target-id6abf26814ebd1\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\">To solve this problem, we must imagine the situation that occurs when the light ray hits the first mirror at an angle relative to the normal <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\in ]\\theta, 180^o[<\/span><\/span>. When this happens, we have a situation like the one shown in the following figure:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-T3bmPeT9SHU\/YSNnIfiDnzI\/AAAAAAAAFdc\/68ireIOvkuYdVvLw0dZD4oqeP2YL8Ml3QCLcBGAsYHQ\/s0\/espejo5.PNG\" width=\"650\" height=\"294\" alt=\"ray against plane mirrors\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-T3bmPeT9SHU\/YSNnIfiDnzI\/AAAAAAAAFdc\/68ireIOvkuYdVvLw0dZD4oqeP2YL8Ml3QCLcBGAsYHQ\/s0\/espejo5.PNG\" width=\"650\" height=\"294\" alt=\"ray against plane mirrors\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">Since the sum of the interior angles of a triangle is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">180^o<\/span><\/span>:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(90^o - \\alpha) + (90^o + \\beta) + \\theta = 180<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Simplifying this relationship, we can obtain the angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> in terms of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta.<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta=\\alpha - \\theta<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">This expression is important because if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta=\\theta,<\/span><\/span> then by the reasoning of the previous exercise, the ray should return on itself in the next reflection. Thus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=2\\theta.<\/span><\/span> Therefore, this reasoning can be extended inductively through:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_0 = 0^o<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_1 = \\theta<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{n-1} = \\alpha_n - \\theta<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">And from this, we have the sequence of return angles:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li style=\"list-style-type: none;\">\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_0 = 0^o<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_1 = \\theta<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{2} = 2\\theta<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/span><\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_{n} = n\\theta<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">In addition, we must note that both the angle between the plane mirrors and each incidence angle must be acute.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Plane Mirrors, Solved Problems Summary: In this class, we will review some solved problems of plane mirrors. The angle of reflection is determined based on the angle between two plane mirrors joined by a hinge, and specific examples are calculated. Critical values of are examined so that the ray bounces once on each mirror, and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27598,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":102,"footnotes":""},"categories":[835,635],"tags":[],"citadela-post-location":[],"class_list":["post-27599","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-geometrical-optics","category-physics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Solved Problems of Plane Mirrors - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Solved Problems of Plane Mirrors: Determine Angles of Reflection, Examine Critical Values with Practical Examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/en\/solved-problems-of-plane-mirrors\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Solved Problems of Plane Mirrors\" \/>\n<meta property=\"og:description\" content=\"Solved Problems of Plane Mirrors: Determine Angles of Reflection, Examine Critical Values with Practical Examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/solved-problems-of-plane-mirrors\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-26T13:00:18+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-07T08:28:57+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/08\/espejosplanos-1024x342.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Solved Problems of Plane Mirrors\" \/>\n<meta name=\"twitter:description\" content=\"Solved Problems of Plane Mirrors: Determine Angles of Reflection, Examine Critical Values with Practical Examples.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/08\/espejosplanos.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Solved Problems of Plane Mirrors\",\"datePublished\":\"2021-08-26T13:00:18+00:00\",\"dateModified\":\"2024-08-07T08:28:57+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/\"},\"wordCount\":1010,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/08\\\/espejosplanos.jpg\",\"articleSection\":[\"Geometrical Optics\",\"Physics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/solved-problems-of-plane-mirrors\\\/\",\"name\":\"Solved Problems of Plane Mirrors - 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