{"id":84934,"date":"2025-06-01T02:56:27","date_gmt":"2025-06-01T02:56:27","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=84934"},"modified":"2025-06-01T02:56:27","modified_gmt":"2025-06-01T02:56:27","slug":"in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar\/","title":{"rendered":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar"},"content":{"rendered":"

In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle around z-axis and \u03b1 denotes the azimuthal angle raised from x-axis. Then the y-component of P\u20d7 is given by<\/p>\n

[amp_mcq option1=”Psin\u03b8sin\u03b1” option2=”Psin\u03b8cos\u03b1” option3=”Pcos\u03b8sin\u03b1” option4=”Pcos\u03b8cos\u03b1” correct=”option1″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CDS-1 – 2019<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nIn spherical polar coordinates (\u03b3, \u03b8, \u03b1), where \u03b3 is the magnitude of the vector P\u20d7 (let’s denote it as P), \u03b8 is the polar angle from the positive z-axis, and \u03b1 is the azimuthal angle from the positive x-axis in the xy-plane, the Cartesian components (Px, Py, Pz) of the vector P\u20d7 are given by:
\n$P_x = P \\sin\\theta \\cos\\alpha$
\n$P_y = P \\sin\\theta \\sin\\alpha$
\n$P_z = P \\cos\\theta$
\nThe question asks for the y-component of P\u20d7. According to the standard conversion from spherical to Cartesian coordinates, the y-component is $P\\sin\\theta\\sin\\alpha$.
\n<\/section>\n
\n– Spherical coordinates typically use (r, \u03b8, \u03c6) or (\u03c1, \u03b8, \u03c6). The question uses (\u03b3, \u03b8, \u03b1) with meanings specified.
\n– \u03b3 (or P) is the magnitude.
\n– \u03b8 is the angle from the z-axis (polar angle).
\n– \u03b1 is the angle from the x-axis in the xy-plane (azimuthal angle).
\n– The projection onto the xy-plane has length $P\\sin\\theta$.
\n– This projection is resolved into x and y components using the azimuthal angle \u03b1.
\n<\/section>\n
\nThe formulas for converting spherical coordinates (P, \u03b8, \u03b1) to Cartesian coordinates (Px, Py, Pz) are derived from trigonometry. The projection of the vector onto the z-axis is $P\\cos\\theta$, giving the z-component. The projection onto the xy-plane has length $P\\sin\\theta$. This projection forms a right triangle in the xy-plane with the x and y axes, where the hypotenuse is $P\\sin\\theta$ and the angle with the x-axis is \u03b1. The x-component is $(P\\sin\\theta)\\cos\\alpha$ and the y-component is $(P\\sin\\theta)\\sin\\alpha$.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle around z-axis and \u03b1 denotes the azimuthal angle raised from x-axis. Then the y-component of P\u20d7 is given by [amp_mcq option1=”Psin\u03b8sin\u03b1” option2=”Psin\u03b8cos\u03b1” option3=”Pcos\u03b8sin\u03b1” option4=”Pcos\u03b8cos\u03b1” correct=”option1″] This question was previously asked in UPSC CDS-1 – 2019 Download PDFAttempt Online In spherical polar coordinates (\u03b3, … <\/p>\n

Detailed SolutionIn spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1087],"tags":[1119,1129,1128],"class_list":["post-84934","post","type-post","status-publish","format-standard","hentry","category-upsc-cds-1","tag-1119","tag-mechanics","tag-physics","no-featured-image-padding"],"yoast_head":"\nIn spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar<\/title>\n<meta name=\"description\" content=\"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), where \u03b3 is the magnitude of the vector P\u20d7 (let's denote it as P), \u03b8 is the polar angle from the positive z-axis, and \u03b1 is the azimuthal angle from the positive x-axis in the xy-plane, the Cartesian components (Px, Py, Pz) of the vector P\u20d7 are given by: $P_x = P sintheta cosalpha$ $P_y = P sintheta sinalpha$ $P_z = P costheta$ The question asks for the y-component of P\u20d7. According to the standard conversion from spherical to Cartesian coordinates, the y-component is $Psinthetasinalpha$. - Spherical coordinates typically use (r, \u03b8, \u03c6) or (\u03c1, \u03b8, \u03c6). The question uses (\u03b3, \u03b8, \u03b1) with meanings specified. - \u03b3 (or P) is the magnitude. - \u03b8 is the angle from the z-axis (polar angle). - \u03b1 is the angle from the x-axis in the xy-plane (azimuthal angle). - The projection onto the xy-plane has length $Psintheta$. - This projection is resolved into x and y components using the azimuthal angle \u03b1.\" \/>\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\/in-spherical-polar-coordinates-\u03b3-\u03b8-\u03b1-\u03b8-denotes-the-polar-angle-ar\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar\" \/>\n<meta property=\"og:description\" content=\"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), where \u03b3 is the magnitude of the vector P\u20d7 (let's denote it as P), \u03b8 is the polar angle from the positive z-axis, and \u03b1 is the azimuthal angle from the positive x-axis in the xy-plane, the Cartesian components (Px, Py, Pz) of the vector P\u20d7 are given by: $P_x = P sintheta cosalpha$ $P_y = P sintheta sinalpha$ $P_z = P costheta$ The question asks for the y-component of P\u20d7. According to the standard conversion from spherical to Cartesian coordinates, the y-component is $Psinthetasinalpha$. - Spherical coordinates typically use (r, \u03b8, \u03c6) or (\u03c1, \u03b8, \u03c6). The question uses (\u03b3, \u03b8, \u03b1) with meanings specified. - \u03b3 (or P) is the magnitude. - \u03b8 is the angle from the z-axis (polar angle). - \u03b1 is the angle from the x-axis in the xy-plane (azimuthal angle). - The projection onto the xy-plane has length $Psintheta$. - This projection is resolved into x and y components using the azimuthal angle \u03b1.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-\u03b3-\u03b8-\u03b1-\u03b8-denotes-the-polar-angle-ar\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T02:56:27+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":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar","description":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), where \u03b3 is the magnitude of the vector P\u20d7 (let's denote it as P), \u03b8 is the polar angle from the positive z-axis, and \u03b1 is the azimuthal angle from the positive x-axis in the xy-plane, the Cartesian components (Px, Py, Pz) of the vector P\u20d7 are given by: $P_x = P sintheta cosalpha$ $P_y = P sintheta sinalpha$ $P_z = P costheta$ The question asks for the y-component of P\u20d7. According to the standard conversion from spherical to Cartesian coordinates, the y-component is $Psinthetasinalpha$. - Spherical coordinates typically use (r, \u03b8, \u03c6) or (\u03c1, \u03b8, \u03c6). The question uses (\u03b3, \u03b8, \u03b1) with meanings specified. - \u03b3 (or P) is the magnitude. - \u03b8 is the angle from the z-axis (polar angle). - \u03b1 is the angle from the x-axis in the xy-plane (azimuthal angle). - The projection onto the xy-plane has length $Psintheta$. - This projection is resolved into x and y components using the azimuthal angle \u03b1.","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\/in-spherical-polar-coordinates-\u03b3-\u03b8-\u03b1-\u03b8-denotes-the-polar-angle-ar\/","og_locale":"en_US","og_type":"article","og_title":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar","og_description":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), where \u03b3 is the magnitude of the vector P\u20d7 (let's denote it as P), \u03b8 is the polar angle from the positive z-axis, and \u03b1 is the azimuthal angle from the positive x-axis in the xy-plane, the Cartesian components (Px, Py, Pz) of the vector P\u20d7 are given by: $P_x = P sintheta cosalpha$ $P_y = P sintheta sinalpha$ $P_z = P costheta$ The question asks for the y-component of P\u20d7. According to the standard conversion from spherical to Cartesian coordinates, the y-component is $Psinthetasinalpha$. - Spherical coordinates typically use (r, \u03b8, \u03c6) or (\u03c1, \u03b8, \u03c6). The question uses (\u03b3, \u03b8, \u03b1) with meanings specified. - \u03b3 (or P) is the magnitude. - \u03b8 is the angle from the z-axis (polar angle). - \u03b1 is the angle from the x-axis in the xy-plane (azimuthal angle). - The projection onto the xy-plane has length $Psintheta$. - This projection is resolved into x and y components using the azimuthal angle \u03b1.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-\u03b3-\u03b8-\u03b1-\u03b8-denotes-the-polar-angle-ar\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T02:56:27+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\/in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar\/","url":"https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar\/","name":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T02:56:27+00:00","dateModified":"2025-06-01T02:56:27+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), where \u03b3 is the magnitude of the vector P\u20d7 (let's denote it as P), \u03b8 is the polar angle from the positive z-axis, and \u03b1 is the azimuthal angle from the positive x-axis in the xy-plane, the Cartesian components (Px, Py, Pz) of the vector P\u20d7 are given by: $P_x = P \\sin\\theta \\cos\\alpha$ $P_y = P \\sin\\theta \\sin\\alpha$ $P_z = P \\cos\\theta$ The question asks for the y-component of P\u20d7. According to the standard conversion from spherical to Cartesian coordinates, the y-component is $P\\sin\\theta\\sin\\alpha$. - Spherical coordinates typically use (r, \u03b8, \u03c6) or (\u03c1, \u03b8, \u03c6). The question uses (\u03b3, \u03b8, \u03b1) with meanings specified. - \u03b3 (or P) is the magnitude. - \u03b8 is the angle from the z-axis (polar angle). - \u03b1 is the angle from the x-axis in the xy-plane (azimuthal angle). - The projection onto the xy-plane has length $P\\sin\\theta$. - This projection is resolved into x and y components using the azimuthal angle \u03b1.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/in-spherical-polar-coordinates-%ce%b3-%ce%b8-%ce%b1-%ce%b8-denotes-the-polar-angle-ar\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CDS-1","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-cds-1\/"},{"@type":"ListItem","position":3,"name":"In spherical polar coordinates (\u03b3, \u03b8, \u03b1), \u03b8 denotes the polar angle ar"}]},{"@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\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?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\/84934","targetHints":{"allow":["GET"]}}],"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=84934"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/84934\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=84934"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=84934"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=84934"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}