{"id":20149,"date":"2024-04-15T05:49:02","date_gmt":"2024-04-15T05:49:02","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20149"},"modified":"2024-04-15T05:49:02","modified_gmt":"2024-04-15T05:49:02","slug":"if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/","title":{"rendered":"If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} &#8211; {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} &#8211; 2{\\rm{z}}} \\right) &#8211; {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a99e571936ec\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    0.3, -2.5, 0.5                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    0.0, 3.0, 2.0                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    0.3, 0.33, 0.5                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"true\">\r\n                    4.0, 3.0, 2.0                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>A vector field is irrotational if its curl is zero. The curl of a vector field is defined as follows:<\/p>\n<p>$$\\nabla \\times \\overrightarrow {\\rm{F}} = \\left( {\\frac{\\partial }{{\\partial {\\rm{x}}}} \\frac{\\partial {\\rm{F}}<em>z}{{\\partial {\\rm{y}}}} &#8211; \\frac{\\partial {\\rm{F}}_y}{{\\partial {\\rm{x}}}} \\right){{\\rm{\\hat a}}<\/em>{\\rm{z}}} + \\left( {\\frac{\\partial }{{\\partial {\\rm{y}}}} \\frac{\\partial {\\rm{F}}<em>x}{{\\partial {\\rm{z}}}} &#8211; \\frac{\\partial {\\rm{F}}_z}{{\\partial {\\rm{y}}}} \\right){{\\rm{\\hat a}}<\/em>{\\rm{x}}} + \\left( <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> {\\frac{\\partial }{{\\partial {\\rm{z}}}} \\frac{\\partial {\\rm{F}}<em>y}{{\\partial {\\rm{x}}}} &#8211; \\frac{\\partial {\\rm{F}}_x}{{\\partial {\\rm{z}}}} \\right){{\\rm{\\hat a}}<\/em>{\\rm{y}}}$$<\/p>\n<p>Substituting the given vector field into the curl operator, we get:<\/p>\n<p>$$\\nabla \\times \\overrightarrow {\\rm{F}} = \\left( {\\frac{\\partial }{{\\partial {\\rm{x}}}} \\frac{\\partial }{{\\partial {\\rm{y}}}} \\left( -{{\\rm{k}}<em>1}{\\rm{z}} \\right) &#8211; \\frac{\\partial }{{\\partial {\\rm{y}}}} \\frac{\\partial }{{\\partial {\\rm{x}}}} \\left( {{\\rm{k}}_2}{\\rm{x}} &#8211; 2{\\rm{z}} \\right) \\right){{\\rm{\\hat a}}<\/em>{\\rm{z}}} + \\left( {\\frac{\\partial }{{\\partial {\\rm{y}}}} \\frac{\\partial }{{\\partial {\\rm{z}}}} \\left( {{\\rm{k}}<em>3}{\\rm{y}} + {\\rm{z}} \\right) &#8211; \\frac{\\partial }{{\\partial {\\rm{z}}}} \\frac{\\partial }{{\\partial {\\rm{y}}}} \\left( 3{\\rm{y}} &#8211; {{\\rm{k}}_1}{\\rm{z}} \\right) \\right){{\\rm{\\hat a}}<\/em>{\\rm{x}}} + \\left( {\\frac{\\partial }{{\\partial {\\rm{z}}}} \\frac{\\partial }{{\\partial {\\rm{x}}}} \\left( -2{\\rm{z}} \\right) &#8211; \\frac{\\partial }{{\\partial {\\rm{x}}}} \\frac{\\partial }{{\\partial {\\rm{z}}}} \\left( {{\\rm{k}}<em>3}{\\rm{y}} + {\\rm{z}} \\right) \\right){{\\rm{\\hat a}}<\/em>{\\rm{y}}}$$<\/p>\n<p>Simplifying, we get:<\/p>\n<p>$$\\nabla \\times \\overrightarrow {\\rm{F}} = \\left( { -{{\\rm{k}}<em>2} + 3{{\\rm{k}}_3}} \\right){{\\rm{\\hat a}}<\/em>{\\rm{x}}} + \\left( { -{{\\rm{k}}<em>3}} \\right){{\\rm{\\hat a}}<\/em>{\\rm{y}}} + \\left( { -{{\\rm{k}}<em>1}} \\right){{\\rm{\\hat a}}<\/em>{\\rm{z}}}$$<\/p>\n<p>For the vector field to be irrotational, the curl must be zero. Therefore, we must have:<\/p>\n<p>$${ -{{\\rm{k}}_2} + 3{{\\rm{k}}_3}} = 0$$<br \/>\n$${ -{{\\rm{k}}_3}} = 0$$<br \/>\n$${ -{{\\rm{k}}_1}} = 0$$<\/p>\n<p>Solving for ${\\rm{k}}_1$, ${\\rm{k}}_2$, and ${\\rm{k}}_3$, we get:<\/p>\n<p>$${\\rm{k}}_1 = 0.3$$<br \/>\n$${\\rm{k}}_2 = 3.0$$<br \/>\n$${\\rm{k}}_3 = 0.5$$<\/p>\n<p>Therefore, the correct answer is $\\boxed{{\\rm{k}}_1 = 0.3, {\\rm{k}}_2 = 3.0, {\\rm{k}}_3 = 0.5}}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} - {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} - 2{\\rm{z}}} \\right) - {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} - {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} - 2{\\rm{z}}} \\right) - {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:49:02+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} - {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} - 2{\\rm{z}}} \\right) - {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/","og_locale":"en_US","og_type":"article","og_title":"If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} - {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} - 2{\\rm{z}}} \\right) - {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:49:02+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/","url":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/","name":"If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} - {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} - 2{\\rm{z}}} \\right) - {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:49:02+00:00","dateModified":"2024-04-15T05:49:02+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/if-the-vector-function-overrightarrow-rmf-rmhat-a_rmxleft-3rmy-rmk_1rmz-right-rmhat-a_rmyleft-rmk_2rmx\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Engineering maths","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/"},{"@type":"ListItem","position":4,"name":"Calculus","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/calculus\/"},{"@type":"ListItem","position":5,"name":"If the vector function \\[\\overrightarrow {\\rm{F}} = {{\\rm{\\hat a}}_{\\rm{x}}}\\left( {3{\\rm{y}} &#8211; {{\\rm{k}}_1}{\\rm{z}}} \\right) + {{\\rm{\\hat a}}_{\\rm{y}}}\\left( {{{\\rm{k}}_2}{\\rm{x}} &#8211; 2{\\rm{z}}} \\right) &#8211; {{\\rm{\\hat a}}_{\\rm{z}}}\\left( {{{\\rm{k}}_3}{\\rm{y}} + {\\rm{z}}} \\right)\\] is irrotational, then the values of the constants k1, k2 and k3 respectively, are A. 0.3, -2.5, 0.5 B. 0.0, 3.0, 2.0 C. 0.3, 0.33, 0.5 D. 4.0, 3.0, 2.0"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20149"}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=20149"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20149\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}