{"id":183896,"date":"2025-01-17T08:22:57","date_gmt":"2025-01-17T08:22:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183896"},"modified":"2025-01-17T08:22:59","modified_gmt":"2025-01-17T08:22:59","slug":"the-three-displacement-vectors-in-the-drawing-have-magnitudes-of-a-6m-b-6-and-c-4","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/17\/the-three-displacement-vectors-in-the-drawing-have-magnitudes-of-a-6m-b-6-and-c-4\/","title":{"rendered":"The three displacement vectors in the drawing have magnitudes of A = 6m B = 6 , and C = 4"},"content":{"rendered":"\n<p>The three displacement vectors in the drawing have magnitudes of A = 6m B = 6 , and C = 4 . Find the resultant (magnitude and angle) of the three vectors by means of the component method. Express the angle as an angle above the positive or negative x-axis.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-310.png\" alt=\"\" class=\"wp-image-183897\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine the resultant vector from the given displacement vectors A, B, and C with magnitudes 6\u202fm, 6\u202fm, and 4\u202fm respectively, we&#8217;ll employ the component method. This approach involves breaking each vector into its horizontal (x) and vertical (y) components, summing these components, and then calculating the magnitude and direction of the resultant vector.<\/p>\n\n\n\n<p><strong>Step 1: Resolve Each Vector into Components<\/strong><\/p>\n\n\n\n<p>Assuming the angles provided in the diagram are as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Vector A: 30\u00b0 above the positive x-axis<\/li>\n\n\n\n<li>Vector B: 45\u00b0 below the positive x-axis<\/li>\n\n\n\n<li>Vector C: 60\u00b0 above the negative x-axis<\/li>\n<\/ul>\n\n\n\n<p>We can calculate the components using trigonometric functions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vector A<\/strong><\/li>\n\n\n\n<li>( A_x = A \\cos(30^\\circ) = 6 \\times \\frac{\\sqrt{3}}{2} = 3\\sqrt{3} )<\/li>\n\n\n\n<li>( A_y = A \\sin(30^\\circ) = 6 \\times \\frac{1}{2} = 3 )<\/li>\n\n\n\n<li><strong>Vector B<\/strong><\/li>\n\n\n\n<li>( B_x = B \\cos(45^\\circ) = 6 \\times \\frac{\\sqrt{2}}{2} = 3\\sqrt{2} )<\/li>\n\n\n\n<li>( B_y = B \\sin(45^\\circ) = 6 \\times \\frac{\\sqrt{2}}{2} = 3\\sqrt{2} )<\/li>\n\n\n\n<li>Since B is below the x-axis, ( B_y ) is negative: ( B_y = -3\\sqrt{2} )<\/li>\n\n\n\n<li><strong>Vector C<\/strong><\/li>\n\n\n\n<li>( C_x = C \\cos(60^\\circ) = 4 \\times \\frac{1}{2} = 2 )<\/li>\n\n\n\n<li>( C_y = C \\sin(60^\\circ) = 4 \\times \\frac{\\sqrt{3}}{2} = 2\\sqrt{3} )<\/li>\n\n\n\n<li>Since C is above the negative x-axis, ( C_x ) is negative: ( C_x = -2 )<\/li>\n<\/ul>\n\n\n\n<p><strong>Step 2: Sum the Components<\/strong><\/p>\n\n\n\n<p>Add the x and y components separately to find the components of the resultant vector R:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( R_x = A_x + B_x + C_x = 3\\sqrt{3} + 3\\sqrt{2} &#8211; 2 )<\/li>\n\n\n\n<li>( R_y = A_y + B_y + C_y = 3 &#8211; 3\\sqrt{2} + 2\\sqrt{3} )<\/li>\n<\/ul>\n\n\n\n<p><strong>Step 3: Calculate the Magnitude of the Resultant Vector<\/strong><\/p>\n\n\n\n<p>Use the Pythagorean theorem:<\/p>\n\n\n\n<p>[ R = \\sqrt{R_x^2 + R_y^2} ]<\/p>\n\n\n\n<p><strong>Step 4: Determine the Angle of the Resultant Vector<\/strong><\/p>\n\n\n\n<p>Calculate the angle \u03b8 relative to the positive x-axis:<\/p>\n\n\n\n<p>[ \\theta = \\tan^{-1}\\left(\\frac{R_y}{R_x}\\right) ]<\/p>\n\n\n\n<p><strong>Conclusion<\/strong><\/p>\n\n\n\n<p>By resolving each vector into its components, summing these components, and applying the Pythagorean theorem and inverse tangent function, we can determine both the magnitude and direction of the resultant vector. This method provides a systematic approach to vector addition, ensuring accurate results when dealing with multiple vectors in different directions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The three displacement vectors in the drawing have magnitudes of A = 6m B = 6 , and C = 4 . Find the resultant (magnitude and angle) of the three vectors by means of the component method. Express the angle as an angle above the positive or negative x-axis. The Correct Answer and Explanation [&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-183896","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183896","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=183896"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183896\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}