{"id":85807,"date":"2025-06-01T03:28:32","date_gmt":"2025-06-01T03:28:32","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=85807"},"modified":"2025-06-01T03:28:32","modified_gmt":"2025-06-01T03:28:32","slug":"the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the\/","title":{"rendered":"The position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The"},"content":{"rendered":"

The position vector of a particle is r\u20d7 = 2t2<\/sup> x\u0302 + 3t y\u0302 + 4 z\u0302. Then the instantaneous velocity v\u20d7 and acceleration a\u20d7 respectively lie<\/p>\n

[amp_mcq option1=”on xy-plane and along z-direction” option2=”on yz-plane and along x-direction” option3=”on yz-plane and along y-direction” option4=”on xy-plane and along x-direction” correct=”option4″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CDS-2 – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe position vector is given by r\u20d7 = 2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302.
\nThe instantaneous velocity vector v\u20d7 is the first derivative of the position vector with respect to time:
\nv\u20d7 = dr\u20d7\/dt = d\/dt (2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302) = (d\/dt 2t\u00b2) x\u0302 + (d\/dt 3t) y\u0302 + (d\/dt 4) z\u0302 = 4t x\u0302 + 3 y\u0302 + 0 z\u0302 = 4t x\u0302 + 3 y\u0302.
\nA vector of the form Ax\u0302 + By\u0302 has components only in the x and y directions. Such a vector lies in the xy-plane.
\nThe instantaneous acceleration vector a\u20d7 is the first derivative of the velocity vector with respect to time:
\na\u20d7 = dv\u20d7\/dt = d\/dt (4t x\u0302 + 3 y\u0302) = (d\/dt 4t) x\u0302 + (d\/dt 3) y\u0302 = 4 x\u0302 + 0 y\u0302 = 4 x\u0302.
\nA vector of the form Ax\u0302 has a component only in the x direction. Such a vector lies along the x-direction.
\nTherefore, the instantaneous velocity v\u20d7 lies on the xy-plane, and the instantaneous acceleration a\u20d7 lies along the x-direction.
\n<\/section>\n
\n– Velocity is the time derivative of the position vector.
\n– Acceleration is the time derivative of the velocity vector.
\n– A vector is in a plane if it has non-zero components only in the directions defining that plane.
\n– A vector is along an axis if it has a non-zero component only in the direction of that axis.
\n<\/section>\n
\nIn this case, the position vector has a constant z-component (4). This means the particle is always located on the plane z=4, which is parallel to the xy-plane. The motion is confined to this plane z=4. The velocity vector (4t x\u0302 + 3 y\u0302) correctly reflects motion in the x and y directions within this plane. The acceleration vector (4 x\u0302) indicates that the acceleration is constant and directed purely in the positive x-direction.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

The position vector of a particle is r\u20d7 = 2t2 x\u0302 + 3t y\u0302 + 4 z\u0302. Then the instantaneous velocity v\u20d7 and acceleration a\u20d7 respectively lie [amp_mcq option1=”on xy-plane and along z-direction” option2=”on yz-plane and along x-direction” option3=”on yz-plane and along y-direction” option4=”on xy-plane and along x-direction” correct=”option4″] This question was previously asked in … <\/p>\n

Detailed SolutionThe position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The<\/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":[1088],"tags":[1114,1129,1128],"class_list":["post-85807","post","type-post","status-publish","format-standard","hentry","category-upsc-cds-2","tag-1114","tag-mechanics","tag-physics","no-featured-image-padding"],"yoast_head":"\nThe position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The<\/title>\n<meta name=\"description\" content=\"The position vector is given by r\u20d7 = 2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302. The instantaneous velocity vector v\u20d7 is the first derivative of the position vector with respect to time: v\u20d7 = dr\u20d7\/dt = d\/dt (2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302) = (d\/dt 2t\u00b2) x\u0302 + (d\/dt 3t) y\u0302 + (d\/dt 4) z\u0302 = 4t x\u0302 + 3 y\u0302 + 0 z\u0302 = 4t x\u0302 + 3 y\u0302. A vector of the form Ax\u0302 + By\u0302 has components only in the x and y directions. Such a vector lies in the xy-plane. The instantaneous acceleration vector a\u20d7 is the first derivative of the velocity vector with respect to time: a\u20d7 = dv\u20d7\/dt = d\/dt (4t x\u0302 + 3 y\u0302) = (d\/dt 4t) x\u0302 + (d\/dt 3) y\u0302 = 4 x\u0302 + 0 y\u0302 = 4 x\u0302. A vector of the form Ax\u0302 has a component only in the x direction. Such a vector lies along the x-direction. Therefore, the instantaneous velocity v\u20d7 lies on the xy-plane, and the instantaneous acceleration a\u20d7 lies along the x-direction. - Velocity is the time derivative of the position vector. - Acceleration is the time derivative of the velocity vector. - A vector is in a plane if it has non-zero components only in the directions defining that plane. - A vector is along an axis if it has a non-zero component only in the direction of that axis.\" \/>\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\/the-position-vector-of-a-particle-is-r\u20d7-2t-2-x\u0302-3t-y-4-\u1e91-the\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The\" \/>\n<meta property=\"og:description\" content=\"The position vector is given by r\u20d7 = 2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302. The instantaneous velocity vector v\u20d7 is the first derivative of the position vector with respect to time: v\u20d7 = dr\u20d7\/dt = d\/dt (2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302) = (d\/dt 2t\u00b2) x\u0302 + (d\/dt 3t) y\u0302 + (d\/dt 4) z\u0302 = 4t x\u0302 + 3 y\u0302 + 0 z\u0302 = 4t x\u0302 + 3 y\u0302. A vector of the form Ax\u0302 + By\u0302 has components only in the x and y directions. Such a vector lies in the xy-plane. The instantaneous acceleration vector a\u20d7 is the first derivative of the velocity vector with respect to time: a\u20d7 = dv\u20d7\/dt = d\/dt (4t x\u0302 + 3 y\u0302) = (d\/dt 4t) x\u0302 + (d\/dt 3) y\u0302 = 4 x\u0302 + 0 y\u0302 = 4 x\u0302. A vector of the form Ax\u0302 has a component only in the x direction. Such a vector lies along the x-direction. Therefore, the instantaneous velocity v\u20d7 lies on the xy-plane, and the instantaneous acceleration a\u20d7 lies along the x-direction. - Velocity is the time derivative of the position vector. - Acceleration is the time derivative of the velocity vector. - A vector is in a plane if it has non-zero components only in the directions defining that plane. - A vector is along an axis if it has a non-zero component only in the direction of that axis.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r\u20d7-2t-2-x\u0302-3t-y-4-\u1e91-the\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T03:28:32+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":"The position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The","description":"The position vector is given by r\u20d7 = 2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302. The instantaneous velocity vector v\u20d7 is the first derivative of the position vector with respect to time: v\u20d7 = dr\u20d7\/dt = d\/dt (2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302) = (d\/dt 2t\u00b2) x\u0302 + (d\/dt 3t) y\u0302 + (d\/dt 4) z\u0302 = 4t x\u0302 + 3 y\u0302 + 0 z\u0302 = 4t x\u0302 + 3 y\u0302. A vector of the form Ax\u0302 + By\u0302 has components only in the x and y directions. Such a vector lies in the xy-plane. The instantaneous acceleration vector a\u20d7 is the first derivative of the velocity vector with respect to time: a\u20d7 = dv\u20d7\/dt = d\/dt (4t x\u0302 + 3 y\u0302) = (d\/dt 4t) x\u0302 + (d\/dt 3) y\u0302 = 4 x\u0302 + 0 y\u0302 = 4 x\u0302. A vector of the form Ax\u0302 has a component only in the x direction. Such a vector lies along the x-direction. Therefore, the instantaneous velocity v\u20d7 lies on the xy-plane, and the instantaneous acceleration a\u20d7 lies along the x-direction. - Velocity is the time derivative of the position vector. - Acceleration is the time derivative of the velocity vector. - A vector is in a plane if it has non-zero components only in the directions defining that plane. - A vector is along an axis if it has a non-zero component only in the direction of that axis.","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\/the-position-vector-of-a-particle-is-r\u20d7-2t-2-x\u0302-3t-y-4-\u1e91-the\/","og_locale":"en_US","og_type":"article","og_title":"The position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The","og_description":"The position vector is given by r\u20d7 = 2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302. The instantaneous velocity vector v\u20d7 is the first derivative of the position vector with respect to time: v\u20d7 = dr\u20d7\/dt = d\/dt (2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302) = (d\/dt 2t\u00b2) x\u0302 + (d\/dt 3t) y\u0302 + (d\/dt 4) z\u0302 = 4t x\u0302 + 3 y\u0302 + 0 z\u0302 = 4t x\u0302 + 3 y\u0302. A vector of the form Ax\u0302 + By\u0302 has components only in the x and y directions. Such a vector lies in the xy-plane. The instantaneous acceleration vector a\u20d7 is the first derivative of the velocity vector with respect to time: a\u20d7 = dv\u20d7\/dt = d\/dt (4t x\u0302 + 3 y\u0302) = (d\/dt 4t) x\u0302 + (d\/dt 3) y\u0302 = 4 x\u0302 + 0 y\u0302 = 4 x\u0302. A vector of the form Ax\u0302 has a component only in the x direction. Such a vector lies along the x-direction. Therefore, the instantaneous velocity v\u20d7 lies on the xy-plane, and the instantaneous acceleration a\u20d7 lies along the x-direction. - Velocity is the time derivative of the position vector. - Acceleration is the time derivative of the velocity vector. - A vector is in a plane if it has non-zero components only in the directions defining that plane. - A vector is along an axis if it has a non-zero component only in the direction of that axis.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r\u20d7-2t-2-x\u0302-3t-y-4-\u1e91-the\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T03:28:32+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\/the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the\/","name":"The position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T03:28:32+00:00","dateModified":"2025-06-01T03:28:32+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The position vector is given by r\u20d7 = 2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302. The instantaneous velocity vector v\u20d7 is the first derivative of the position vector with respect to time: v\u20d7 = dr\u20d7\/dt = d\/dt (2t\u00b2 x\u0302 + 3t y\u0302 + 4 z\u0302) = (d\/dt 2t\u00b2) x\u0302 + (d\/dt 3t) y\u0302 + (d\/dt 4) z\u0302 = 4t x\u0302 + 3 y\u0302 + 0 z\u0302 = 4t x\u0302 + 3 y\u0302. A vector of the form Ax\u0302 + By\u0302 has components only in the x and y directions. Such a vector lies in the xy-plane. The instantaneous acceleration vector a\u20d7 is the first derivative of the velocity vector with respect to time: a\u20d7 = dv\u20d7\/dt = d\/dt (4t x\u0302 + 3 y\u0302) = (d\/dt 4t) x\u0302 + (d\/dt 3) y\u0302 = 4 x\u0302 + 0 y\u0302 = 4 x\u0302. A vector of the form Ax\u0302 has a component only in the x direction. Such a vector lies along the x-direction. Therefore, the instantaneous velocity v\u20d7 lies on the xy-plane, and the instantaneous acceleration a\u20d7 lies along the x-direction. - Velocity is the time derivative of the position vector. - Acceleration is the time derivative of the velocity vector. - A vector is in a plane if it has non-zero components only in the directions defining that plane. - A vector is along an axis if it has a non-zero component only in the direction of that axis.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-position-vector-of-a-particle-is-r%e2%83%97-2t-2-x%cc%82-3t-y-4-%e1%ba%91-the\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CDS-2","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-cds-2\/"},{"@type":"ListItem","position":3,"name":"The position vector of a particle is r\u20d7 = 2t 2 x\u0302 + 3t y\u0302 + 4 z\u0302. The"}]},{"@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\/85807","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=85807"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/85807\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=85807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=85807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=85807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}