{"id":250320,"date":"2025-07-10T08:48:13","date_gmt":"2025-07-10T08:48:13","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=250320"},"modified":"2025-07-10T08:48:15","modified_gmt":"2025-07-10T08:48:15","slug":"find-the-resultant-vector-magnitude-and-direction-by-adding-the-following-set-of-vectors","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/find-the-resultant-vector-magnitude-and-direction-by-adding-the-following-set-of-vectors\/","title":{"rendered":"Find the resultant vector (magnitude and direction) by adding the following set of vectors"},"content":{"rendered":"\n

Find the resultant vector (magnitude and direction) by adding the following set of vectors: A – 7.7m due South B = 4.5m at 60\u00c2\u00b0 South of East C = 7.4m due East State the direction as an angle between 0 and 90 degrees north of east, north of west, south of east, or south of west.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find the resultant vector when adding vectors A, B, and C, we need to break each vector into its components along the x-axis (East-West) and y-axis (North-South). Then, we can sum up the components in each direction to get the resultant vector. Finally, we’ll find its magnitude and direction.<\/p>\n\n\n\n

Step 1: Resolve each vector into components<\/h3>\n\n\n\n

Vector A (7.7 m due South):<\/h4>\n\n\n\n
    \n
  • x-component: 0 (since it only points south, there is no East-West component)<\/li>\n\n\n\n
  • y-component: -7.7 m (negative because it points south)<\/li>\n<\/ul>\n\n\n\n

    Vector B (4.5 m at 60\u00b0 South of East):<\/h4>\n\n\n\n
      \n
    • x-component: Bx=4.5\u00d7cos\u2061(60\u2218)=4.5\u00d70.5=2.25B_x = 4.5 \\times \\cos(60^\\circ) = 4.5 \\times 0.5 = 2.25Bx\u200b=4.5\u00d7cos(60\u2218)=4.5\u00d70.5=2.25 m (eastward)<\/li>\n\n\n\n
    • y-component: By=4.5\u00d7sin\u2061(60\u2218)=4.5\u00d70.866=3.897B_y = 4.5 \\times \\sin(60^\\circ) = 4.5 \\times 0.866 = 3.897By\u200b=4.5\u00d7sin(60\u2218)=4.5\u00d70.866=3.897 m (southward)<\/li>\n<\/ul>\n\n\n\n

      Vector C (7.4 m due East):<\/h4>\n\n\n\n
        \n
      • x-component: 7.4 m (eastward)<\/li>\n\n\n\n
      • y-component: 0 (since it only points east, there is no North-South component)<\/li>\n<\/ul>\n\n\n\n

        Step 2: Sum up the components<\/h3>\n\n\n\n

        x-components (East-West):<\/h4>\n\n\n\n

        Rx=Ax+Bx+Cx=0+2.25+7.4=9.65\u2009mR_x = A_x + B_x + C_x = 0 + 2.25 + 7.4 = 9.65 \\, \\text{m}Rx\u200b=Ax\u200b+Bx\u200b+Cx\u200b=0+2.25+7.4=9.65m<\/p>\n\n\n\n

        y-components (North-South):<\/h4>\n\n\n\n

        Ry=Ay+By+Cy=\u22127.7+(\u22123.897)+0=\u221211.597\u2009mR_y = A_y + B_y + C_y = -7.7 + (-3.897) + 0 = -11.597 \\, \\text{m}Ry\u200b=Ay\u200b+By\u200b+Cy\u200b=\u22127.7+(\u22123.897)+0=\u221211.597m<\/p>\n\n\n\n

        Step 3: Find the magnitude of the resultant vector<\/h3>\n\n\n\n

        The magnitude of the resultant vector RRR is given by the Pythagorean theorem: R=Rx2+Ry2=(9.65)2+(\u221211.597)2R = \\sqrt{R_x^2 + R_y^2} = \\sqrt{(9.65)^2 + (-11.597)^2}R=Rx2\u200b+Ry2\u200b\u200b=(9.65)2+(\u221211.597)2\u200b R=93.1225+134.4328=227.5553\u224815.08\u2009mR = \\sqrt{93.1225 + 134.4328} = \\sqrt{227.5553} \\approx 15.08 \\, \\text{m}R=93.1225+134.4328\u200b=227.5553\u200b\u224815.08m<\/p>\n\n\n\n

        Step 4: Find the direction of the resultant vector<\/h3>\n\n\n\n

        The direction is the angle \u03b8\\theta\u03b8 between the resultant vector and the positive x-axis (East), measured counterclockwise. We can use the inverse tangent function: \u03b8=tan\u2061\u22121(RyRx)=tan\u2061\u22121(\u221211.5979.65)\\theta = \\tan^{-1}\\left(\\frac{R_y}{R_x}\\right) = \\tan^{-1}\\left(\\frac{-11.597}{9.65}\\right)\u03b8=tan\u22121(Rx\u200bRy\u200b\u200b)=tan\u22121(9.65\u221211.597\u200b) \u03b8=tan\u2061\u22121(\u22121.20)\u2248\u221250.19\u2218\\theta = \\tan^{-1}(-1.20) \\approx -50.19^\\circ\u03b8=tan\u22121(\u22121.20)\u2248\u221250.19\u2218<\/p>\n\n\n\n

        Since the angle is negative, it means the vector is pointing south of east<\/strong>. To express the angle as a positive value: \u03b8=50.19\u2218\u2009south of east\\theta = 50.19^\\circ \\, \\text{south of east}\u03b8=50.19\u2218south of east<\/p>\n\n\n\n

        Final Answer:<\/h3>\n\n\n\n
          \n
        • Magnitude: 15.08 m<\/li>\n\n\n\n
        • Direction: 50.19\u00b0 south of east<\/li>\n<\/ul>\n\n\n\n
          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          Find the resultant vector (magnitude and direction) by adding the following set of vectors: A – 7.7m due South B = 4.5m at 60\u00c2\u00b0 South of East C = 7.4m due East State the direction as an angle between 0 and 90 degrees north of east, north of west, south of east, or south of […]<\/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-250320","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250320","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=250320"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250320\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=250320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=250320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=250320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}