{"id":255907,"date":"2025-07-13T11:15:21","date_gmt":"2025-07-13T11:15:21","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=255907"},"modified":"2025-07-13T11:15:24","modified_gmt":"2025-07-13T11:15:24","slug":"specify-the-magnitude-f3-and-directions-%ce%b1%e2%82%83-%ce%b2%e2%82%83-and-of-so-that-the-force-of-the-three-forces","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/13\/specify-the-magnitude-f3-and-directions-%ce%b1%e2%82%83-%ce%b2%e2%82%83-and-of-so-that-the-force-of-the-three-forces\/","title":{"rendered":"Specify the magnitude F3\u00a0\u00a0and directions\u00a0\u03b1\u2083 ,\u00a0,\u03b2\u2083 and\u00a0\u00a0of\u00a0\u00a0so that the force of the three forces"},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">Specify the magnitude F3\u00a0\u00a0and directions\u00a0<strong>\u03b1\u2083 <\/strong>,\u00a0,<strong>\u03b2\u2083 <\/strong> and\u00a0\u00a0of\u00a0\u00a0so that the force of the three forces is\u00a0<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-462.png\" alt=\"\" class=\"wp-image-255908\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong><br>The magnitude and direction angles of the force F\u2083 are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>F\u2083 = 10.6 kN<\/strong><\/li>\n\n\n\n<li><strong>\u03b1\u2083 = 125\u00b0<\/strong><\/li>\n\n\n\n<li><strong>\u03b2\u2083 = 42.3\u00b0<\/strong><\/li>\n\n\n\n<li><strong>\u03b3\u2083 = 111\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>To determine the properties of force F\u2083, we use the principle of vector addition. The resultant force, F_R, is the vector sum of the three individual forces: F_R = F\u2081 + F\u2082 + F\u2083. The problem requires this resultant to be F_R = {9j} kN.<\/p>\n\n\n\n<p>First, we must express each force in its Cartesian vector components (i, j, k).<\/p>\n\n\n\n<p>Force F\u2081 has a magnitude of 12 kN and lies in the x-y plane. Its components are found using trigonometry, with the angle 30\u00b0 given with respect to the y-axis.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>F\u2081\u2093 = 12 sin(30\u00b0) = 6 kN<\/li>\n\n\n\n<li>F\u2081\u1d67 = 12 cos(30\u00b0) = 10.39 kN<\/li>\n\n\n\n<li>F\u2081 = {6i + 10.39j} kN<\/li>\n<\/ul>\n\n\n\n<p>Force F\u2082 has a magnitude of 10 kN and lies in the y-z plane. Its components are determined using the provided 5-12-13 slope triangle. It acts in the negative y and positive z directions.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>F\u2082\u1d67 = -10 * (12\/13) = -9.23 kN<\/li>\n\n\n\n<li>F\u2082z = 10 * (5\/13) = 3.85 kN<\/li>\n\n\n\n<li>F\u2082 = {-9.23j + 3.85k} kN<\/li>\n<\/ul>\n\n\n\n<p>The unknown force is F\u2083 = {F\u2083\u2093 i + F\u2083\u1d67 j + F\u2083z k}.<\/p>\n\n\n\n<p>Since the sum of the forces equals the resultant, we can equate the sum of the components on each axis:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>x-axis:<\/strong>\u00a00 = F\u2081\u2093 + F\u2082\u2093 + F\u2083\u2093 \u2192 0 = 6 + 0 + F\u2083\u2093 \u2192\u00a0<strong>F\u2083\u2093 = -6 kN<\/strong><\/li>\n\n\n\n<li><strong>y-axis:<\/strong>\u00a09 = F\u2081\u1d67 + F\u2082\u1d67 + F\u2083\u1d67 \u2192 9 = 10.39 &#8211; 9.23 + F\u2083\u1d67 \u2192\u00a0<strong>F\u2083\u1d67 = 7.84 kN<\/strong><\/li>\n\n\n\n<li><strong>z-axis:<\/strong>\u00a00 = F\u2081z + F\u2082z + F\u2083z \u2192 0 = 0 + 3.85 + F\u2083z \u2192\u00a0<strong>F\u2083z = -3.85 kN<\/strong><\/li>\n<\/ul>\n\n\n\n<p>With the components of F\u2083 known, we find its magnitude using the 3D Pythagorean theorem:<br>F\u2083 = \u221a(F\u2083\u2093\u00b2 + F\u2083\u1d67\u00b2 + F\u2083z\u00b2) = \u221a((-6)\u00b2 + (7.84)\u00b2 + (-3.85)\u00b2) \u2248&nbsp;<strong>10.6 kN<\/strong>.<\/p>\n\n\n\n<p>The direction angles are then found using the inverse cosine of each component divided by the magnitude:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03b1\u2083 = arccos(F\u2083\u2093 \/ F\u2083) = arccos(-6 \/ 10.6) \u2248\u00a0<strong>125\u00b0<\/strong><\/li>\n\n\n\n<li>\u03b2\u2083 = arccos(F\u2083\u1d67 \/ F\u2083) = arccos(7.84 \/ 10.6) \u2248\u00a0<strong>42.3\u00b0<\/strong><\/li>\n\n\n\n<li>\u03b3\u2083 = arccos(F\u2083z \/ F\u2083) = arccos(-3.85 \/ 10.6) \u2248\u00a0<strong>111\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-521.jpeg\" alt=\"\" class=\"wp-image-255909\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Specify the magnitude F3\u00a0\u00a0and directions\u00a0\u03b1\u2083 ,\u00a0,\u03b2\u2083 and\u00a0\u00a0of\u00a0\u00a0so that the force of the three forces is\u00a0 The Correct Answer and Explanation is: Correct Answer:The magnitude and direction angles of the force F\u2083 are: Explanation: To determine the properties of force F\u2083, we use the principle of vector addition. The resultant force, F_R, is the vector sum [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-255907","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/255907","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=255907"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/255907\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=255907"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=255907"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=255907"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}