{"id":28215,"date":"2025-06-20T11:02:00","date_gmt":"2025-06-20T11:02:00","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=28215"},"modified":"2025-06-20T11:02:02","modified_gmt":"2025-06-20T11:02:02","slug":"determine-the-magnitude-and-coordinate-direction-angles-of-the-resultant-force-and-sketch-this-vector-on-the-coordinate-system","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/determine-the-magnitude-and-coordinate-direction-angles-of-the-resultant-force-and-sketch-this-vector-on-the-coordinate-system\/","title":{"rendered":"Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-black-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To determine the <strong>magnitude<\/strong> and <strong>coordinate direction angles<\/strong> of the resultant force, we first need the vector components of all forces involved. Since you haven&#8217;t provided a specific diagram or force vectors, I\u2019ll explain the general method with a common example and then show how to sketch it.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Example:<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose we are given three forces acting at a point in space:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>F\u2081<\/strong> = 100 N along the x-axis \u2192 F\u2081 = \u27e8100, 0, 0\u27e9<\/li>\n\n\n\n<li><strong>F\u2082<\/strong> = 80 N in the y-direction \u2192 F\u2082 = \u27e80, 80, 0\u27e9<\/li>\n\n\n\n<li><strong>F\u2083<\/strong> = 60 N in the z-direction \u2192 F\u2083 = \u27e80, 0, 60\u27e9<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 1: Add the Force Components<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The resultant force <strong>FR<\/strong> is the sum of the vectors: FR=F1+F2+F3=\u27e8100,0,0\u27e9+\u27e80,80,0\u27e9+\u27e80,0,60\u27e9=\u27e8100,80,60\u27e9FR = F\u2081 + F\u2082 + F\u2083 = \u27e8100, 0, 0\u27e9 + \u27e80, 80, 0\u27e9 + \u27e80, 0, 60\u27e9 = \u27e8100, 80, 60\u27e9FR=F1\u200b+F2\u200b+F3\u200b=\u27e8100,0,0\u27e9+\u27e80,80,0\u27e9+\u27e80,0,60\u27e9=\u27e8100,80,60\u27e9<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 2: Find the Magnitude of the Resultant<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Use the 3D magnitude formula: \u2223FR\u2223=(100)2+(80)2+(60)2=10000+6400+3600=20000=141.42\u2009N|FR| = \\sqrt{(100)^2 + (80)^2 + (60)^2} = \\sqrt{10000 + 6400 + 3600} = \\sqrt{20000} = 141.42 \\, \\text{N}\u2223FR\u2223=(100)2+(80)2+(60)2\u200b=10000+6400+3600\u200b=20000\u200b=141.42N<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 3: Coordinate Direction Angles<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Use cosine relations: cos\u2061(\u03b1)=FRx\u2223FR\u2223=100141.42=0.707\u21d2\u03b1=cos\u2061\u22121(0.707)=45\u2218\\cos(\\alpha) = \\frac{FR_x}{|FR|} = \\frac{100}{141.42} = 0.707 \\Rightarrow \\alpha = \\cos^{-1}(0.707) = 45^\\circcos(\u03b1)=\u2223FR\u2223FRx\u200b\u200b=141.42100\u200b=0.707\u21d2\u03b1=cos\u22121(0.707)=45\u2218 cos\u2061(\u03b2)=FRy\u2223FR\u2223=80141.42=0.566\u21d2\u03b2=cos\u2061\u22121(0.566)=55.1\u2218\\cos(\\beta) = \\frac{FR_y}{|FR|} = \\frac{80}{141.42} = 0.566 \\Rightarrow \\beta = \\cos^{-1}(0.566) = 55.1^\\circcos(\u03b2)=\u2223FR\u2223FRy\u200b\u200b=141.4280\u200b=0.566\u21d2\u03b2=cos\u22121(0.566)=55.1\u2218 cos\u2061(\u03b3)=FRz\u2223FR\u2223=60141.42=0.424\u21d2\u03b3=cos\u2061\u22121(0.424)=64.9\u2218\\cos(\\gamma) = \\frac{FR_z}{|FR|} = \\frac{60}{141.42} = 0.424 \\Rightarrow \\gamma = \\cos^{-1}(0.424) = 64.9^\\circcos(\u03b3)=\u2223FR\u2223FRz\u200b\u200b=141.4260\u200b=0.424\u21d2\u03b3=cos\u22121(0.424)=64.9\u2218<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Sketching the Vector<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Draw a 3D coordinate system with x, y, and z axes. From the origin, draw a vector pointing into the first octant with components 100 in x, 80 in y, and 60 in z. This vector represents <strong>FR<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Summary<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Resultant Force Magnitude<\/strong>: 141.42 N<\/li>\n\n\n\n<li><strong>Direction Angles<\/strong>:\n<ul class=\"wp-block-list\">\n<li>\u03b1 = 45\u00b0 with x-axis<\/li>\n\n\n\n<li>\u03b2 = 55.1\u00b0 with y-axis<\/li>\n\n\n\n<li>\u03b3 = 64.9\u00b0 with z-axis<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-112.jpeg\" alt=\"\" class=\"wp-image-28223\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-112.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-112-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-112-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-112-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system. The Correct Answer and Explanation is: To determine the magnitude and coordinate direction angles of the resultant force, we first need the vector components of all forces involved. Since you haven&#8217;t provided a specific diagram or [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-28215","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28215","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=28215"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28215\/revisions"}],"predecessor-version":[{"id":28226,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/28215\/revisions\/28226"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=28215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=28215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=28215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}