{"id":2044,"date":"2018-04-22T18:23:52","date_gmt":"2018-04-22T10:23:52","guid":{"rendered":"https:\/\/gurumuda.net\/physics\/?p=2044"},"modified":"2023-08-09T04:23:50","modified_gmt":"2023-08-09T04:23:50","slug":"adding-vectors-using-components-problems-and-solutions","status":"publish","type":"post","link":"https:\/\/gurumuda.net\/physics\/adding-vectors-using-components-problems-and-solutions.htm","title":{"rendered":"Adding vectors using components \u2013 problems and solutions","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

Adding vectors using components \u2013 problems and solutions<\/span><\/p>\n

1. Three vectors as shown in the figure below.<\/span><\/p>\n

V1 <\/sub> = 30\"Adding<\/span><\/p>\n

V2<\/sub> = 30<\/span><\/p>\n

V3 <\/sub> = 40<\/span><\/p>\n

What is the resultant vectors<\/a>.<\/span><\/p>\n

Known :<\/span><\/p>\n

V1<\/sub> = 30, angle between V1<\/sub> and x axis = 30o<\/sup><\/span><\/p>\n

V2<\/sub> = 30, angle between V2<\/sub> and x axis = 30o<\/sup><\/span><\/p>\n

V3<\/sub> = 40, angle between V3 <\/sub>and x axis = 0o<\/sup><\/span><\/p>\n

Wanted :<\/u> The resultant vectors<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

The components of vectors<\/a> :<\/u><\/span><\/p>\n

V1x<\/sub> = (V1<\/sub>)(cos 30o<\/sup>) = (30)(0.5\u221a3) = 15\u221a3. Positive because this vector component points along the positive x axis (rightward).<\/span><\/p>\n

V1y<\/sub> = (V1<\/sub>)(sin 30o<\/sup>) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).<\/span><\/p>\n

V2x<\/sub> = (V2<\/sub>)(cos 30o<\/sup>) = (30)(0.5\u221a3) = -15\u221a3. Negative because this vector component points along the negative x axis (leftward).<\/span><\/p>\n

V2y<\/sub> = (V2<\/sub>)(sin 30o<\/sup>) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).<\/span><\/p>\n

V3x<\/sub> = (V3<\/sub>)(cos 0o<\/sup>) = (40)(1) = 40. Positive because this vector component points along the positive x axis (rightward).<\/span><\/p>\n

V3y<\/sub> = (V3<\/sub>)(sin 0o<\/sup>) = (40)(0) = 0<\/span><\/p>\n

The components of the resultant vectors :<\/u><\/span><\/p>\n

Vx<\/sub> = V1x <\/sub>– V2x<\/sub> + V3x <\/sub>= 15\u221a3 \u2013 15\u221a3 + 40 = 40 <\/span><\/p>\n

Vy<\/sub> = V1y <\/sub>+ V2y <\/sub>+ V3y <\/sub>= 15 + 15 = 30 <\/span><\/p>\n

The resultant vector<\/a> :<\/u><\/span><\/p>\n

\"Adding<\/span><\/p>\n

2. Two forces perpendicular each other, F1 <\/sub>= 12 N and F2<\/sub> = 5 N. What is the resultant of both forces.<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

Force <\/span>1 (F<\/span>1<\/span><\/sub>) = 12 Newton<\/span><\/span><\/p>\n

Force <\/span>2 (F<\/span>2<\/span><\/sub>) = 5 Newton <\/span><\/span><\/p>\n

Wanted : <\/u>The resultant vectors <\/span>(<\/span>\u03a3F)<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

\u03a3F<\/span>2<\/span><\/sup> = F<\/span>1<\/span><\/sub>2<\/span><\/sup> + F<\/span>2<\/span><\/sub>2 <\/span><\/sup>= <\/span>12<\/span>2 <\/span><\/sup>+ <\/span>5<\/span>2 <\/span><\/sup>= 144 + 25 = 169<\/span><\/span><\/p>\n

\u03a3F = <\/span>\u221a<\/span>169 = 13 Newton<\/span><\/span><\/p>\n

3. Three vectors,<\/span><\/p>\n

V1 <\/sub>= 30\"Adding<\/span><\/p>\n

V2 <\/sub>= 30<\/span><\/p>\n

V3<\/sub> = 40<\/span><\/p>\n

Determine the resultant vectors.<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

v<\/span>1<\/span><\/sub> = 30, <\/span>makes <\/span>30<\/span>o<\/span><\/sup> about the negative x axis<\/span><\/span><\/p>\n

v<\/span>2<\/span><\/sub> = 30, <\/span>makes <\/span>30<\/span>o<\/span><\/sup> about the positive x axis<\/span><\/span><\/p>\n

v<\/span>3<\/span><\/sub> = 40, <\/span>makes <\/span>0<\/span>o<\/span><\/sup> about the positive x axis<\/span><\/span><\/p>\n

Wanted :<\/u> The resultant vectors<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

The components of vectors :<\/u><\/span><\/p>\n

v<\/span>1x<\/span><\/sub> = v<\/span>1 <\/span><\/sub>cos 30<\/span>o<\/span><\/sup> = (30)(0.5<\/span>\u221a<\/span>3) = -15<\/span>\u221a<\/span>3 (<\/span>Negative because this vector component points along the negative x axis (leftward)<\/span>)<\/span><\/span><\/p>\n

v<\/span>1y<\/span><\/sub> = v<\/span>1<\/span><\/sub> sin 30<\/span>o<\/span><\/sup> = (30)(0.5) = 15 (<\/span>Positive because this vector component points along the positive y axis (upward)<\/span>)<\/span><\/span><\/p>\n

v<\/span>2x<\/span><\/sub> = v<\/span>2 <\/span><\/sub>cos 30<\/span>o<\/span><\/sup> = (30)(0.5<\/span>\u221a<\/span>3) = 15<\/span>\u221a<\/span>3 (<\/span>Positive because this vector component points along the positive x axis (rightward<\/span>)<\/span>)<\/span><\/span><\/p>\n

v<\/span>2y<\/span><\/sub> = v<\/span>2 <\/span><\/sub>sin 30<\/span>o<\/span><\/sup> = (30)(0.5) = 15 (<\/span>Positive because this vector component points along the positive y axis (upward).<\/span>)<\/span><\/span><\/p>\n

v<\/span>3x <\/span><\/sub>= v<\/span>3 <\/span><\/sub>cos 0<\/span>o<\/span><\/sup> = (40)(1) = 40 (<\/span>Positive because this vector component points along the positive x axis (rightward).<\/span>)<\/span><\/span><\/p>\n

v<\/span>3y <\/span><\/sub>= v<\/span>3<\/span><\/sub> sin 0<\/span>o<\/span><\/sup> = (40)(0) = 0<\/span><\/span><\/p>\n

The components of the resultant vectors :<\/u><\/span><\/p>\n

v<\/span>x<\/span><\/sub> = – v<\/span>1x<\/span><\/sub> + v<\/span>2x<\/span><\/sub> + v<\/span>3x<\/span><\/sub> = -15<\/span>\u221a<\/span>3 + 15<\/span>\u221a<\/span>3 + 40 = 40<\/span><\/span><\/p>\n

v<\/span>y<\/span><\/sub> = v<\/span>1y<\/span><\/sub> + v<\/span>2y<\/span><\/sub> + v<\/span>3y<\/span><\/sub> = 15 + 15 = 30<\/span><\/span><\/p>\n

The resultant vector :<\/u><\/span><\/p>\n

\"Adding<\/span><\/p>\n

4<\/span>. <\/span>What is the resultant of three vectors as shown in figure below :<\/span><\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

F<\/span>1<\/span><\/sub> = 3 Newton, <\/span>makes <\/span>60<\/span>o<\/span><\/sup> about the positive x axis\"Adding<\/span><\/span><\/p>\n

F<\/span>2<\/span><\/sub> = 3 Newton, <\/span>makes <\/span>0<\/span>o<\/span><\/sup> about the negative x axis <\/span><\/span><\/p>\n

F<\/span>3<\/span><\/sub> = 6 Newto<\/span>n, makes <\/span>60<\/span>o<\/span><\/sup> about the negative y axis<\/span><\/span><\/p>\n

Wanted :<\/u> The resultant vector<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

The components of vectors :<\/u><\/span><\/p>\n

F1x<\/sub> = F1<\/sub> cos 60o<\/sup> = (3)(0.5) = 1.5 N (Positive because this vector component points along the positive x axis (rightward<\/span>)<\/span>)<\/span><\/p>\n

F1y<\/sub> = F1<\/sub> sin 60o<\/sup> = (3)(0.5\u221a3) = 1.5\u221a3 N (Positive because this vector component points along the positive y axis (upward)<\/span>)<\/span><\/p>\n

F2x<\/sub> = F2<\/sub> cos 0o<\/sup> = (3)(1) = -3 N (Negative because this vector component points along the negative x axis (leftward)<\/span>)<\/span><\/p>\n

F2y<\/sub> = F2<\/sub> sin 0o<\/sup> = (3)(0) = 0<\/span><\/p>\n

F3x<\/sub> = F3<\/sub> cos 60o<\/sup> = (6)(0.5) = 3 N (Positive because this vector component points along the positive x axis (rightward<\/span>)<\/span>)<\/span><\/p>\n

F3y<\/sub> = F3<\/sub> sin 60o<\/sup> = (6)(0.5\u221a3) = -3\u221a3 N (Negative because this vector component points along the negative <\/span>y<\/span> axis (<\/span>down<\/span>ward)<\/span>)<\/span><\/p>\n

The components of the resultant vectors :<\/u><\/span><\/p>\n

\u03a3Fx<\/sub> = F1x<\/sub> \u2013 F2x<\/sub> + F3x <\/sub>= 1.5 N \u2013 3 N + 3 N = 1.5 N<\/span><\/p>\n

\u03a3Fy<\/sub> = F1y<\/sub> + F2y<\/sub> – F3y <\/sub>= 1.5\u221a3 N + 0 N – 3\u221a3 N = -1.5\u221a3 N<\/span><\/p>\n

The resultant vector :<\/span><\/p>\n

\"Adding<\/span><\/p>\n

5<\/span>. <\/span>Two forces,<\/span> F1 <\/sub>= 15 N and F2<\/sub> = 9 N. The angle between both vectors is 60\u00b0. What is the resultant of vectors.<\/span><\/p>\n

Wanted :<\/u><\/span><\/p>\n

Force 1 <\/span>(F<\/span>1<\/span><\/sub>) = 15 Newton<\/span><\/span><\/p>\n

Force <\/span>2 (F<\/span>2<\/span><\/sub>) = 9 Newton<\/span><\/span><\/p>\n

Angle <\/span>(<\/span>\u03b8) = 60<\/span>o<\/span><\/sup><\/span><\/p>\n

Wanted:<\/u> The resultant vector<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

\"Adding<\/span><\/p>\n

6. What is the resultant of three vectors as shown in the figure below?<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

F1<\/sub> = 20 Newton, angle between F1 <\/sub>and x axis = 0\"Adding<\/span><\/p>\n

F2<\/sub> = 20 Newton, angle between F2 <\/sub>and x axis = 60<\/span><\/p>\n

F3<\/sub> = 24 Newton, angle between F3 <\/sub>and x axis = 60<\/span><\/p>\n

Wanted :<\/u> The resultant vector<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

The components of vectors :<\/u><\/span><\/p>\n

F1x<\/sub> = (F1<\/sub>)(cos 0) = (20)(1) = 20. Positive because this vector component points along the positive x axis (rightward<\/span>)<\/span><\/span><\/p>\n

F1y<\/sub> = (F1<\/sub>)(sin 0) = (20)(0) = 0<\/span><\/p>\n

F2x<\/sub> = (F2<\/sub>)(cos 60) = (20)(0.5) = -10. Negative because this vector component points along the negative x axis (leftward)<\/span><\/span><\/p>\n

F2y<\/sub> = (F2<\/sub>)(sin 60) = (20)(0.5\u221a3) = 10\u221a3. Positive because this vector component points along the positive y axis (upward)<\/span><\/span><\/p>\n

F3x<\/sub> = (F3<\/sub>)(cos 60) = (24)(0.5) = -12. Negative because this vector component points along the negative x axis (leftward)<\/span><\/span><\/p>\n

F3y<\/sub> = (F3<\/sub>)(sin 60) = (24)(0.5\u221a3) = -12\u221a3. Negative because this vector component points along the negative <\/span>y<\/span> axis (<\/span>down<\/span>ward)<\/span><\/span><\/p>\n

The components of the resultant vectors :<\/u><\/span><\/p>\n

Fx<\/sub> = F1x <\/sub>– F2x <\/sub>– F3x <\/sub>= 20 \u2013 10 \u2013 12 = -2 <\/span><\/p>\n

Fy<\/sub> = F1y <\/sub>+ F2y <\/sub>\u2013 F3y <\/sub>= 0 + 10\u221a3 – 12\u221a3 = -2\u221a3 <\/span><\/p>\n

The resultant vector :<\/u><\/span><\/p>\n

\"Adding<\/span><\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

Adding vectors using components \u2013 problems and solutions 1. Three vectors as shown in the figure below. V1 = 30 V2 = 30 V3 = 40 What is the resultant vectors. Known : V1 = 30, angle between V1 and x axis = 30o<\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_seopress_titles_title":"","_seopress_titles_desc":"","_seopress_robots_index":"","_seopress_robots_follow":"","_seopress_robots_imageindex":"","_seopress_robots_snippet":"","_seopress_robots_primary_cat":"","_seopress_robots_breadcrumbs":"","_seopress_robots_freeze_modified_date":"","_seopress_robots_custom_modified_date":"","_seopress_robots_canonical":"","_seopress_social_fb_title":"","_seopress_social_fb_desc":"","_seopress_social_fb_img":"","_seopress_social_fb_img_attachment_id":0,"_seopress_social_fb_img_width":0,"_seopress_social_fb_img_height":0,"_seopress_social_twitter_title":"","_seopress_social_twitter_desc":"","_seopress_social_twitter_img":"","_seopress_social_twitter_img_attachment_id":0,"_seopress_social_twitter_img_width":0,"_seopress_social_twitter_img_height":0,"_seopress_redirections_value":"","_seopress_redirections_enabled":"","_seopress_redirections_enabled_regex":"","_seopress_redirections_logged_status":"","_seopress_redirections_param":"","_seopress_redirections_type":0,"_seopress_analysis_target_kw":"Adding vectors using components \u2013 problems and solutions","_seopress_news_disabled":"","_seopress_video_disabled":"","_seopress_video":[],"_seopress_pro_schemas_manual":[],"_seopress_pro_rich_snippets_disable_all":"","_seopress_pro_rich_snippets_disable":[],"_seopress_pro_schemas":[],"footnotes":""},"categories":[3],"tags":[],"class_list":["post-2044","post","type-post","status-publish","format-standard","hentry","category-solved-problems-in-basic-physics"],"gt_translate_keys":[{"key":"link","format":"url"}],"_links":{"self":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2044","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/comments?post=2044"}],"version-history":[{"count":2,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2044\/revisions"}],"predecessor-version":[{"id":8659,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2044\/revisions\/8659"}],"wp:attachment":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/media?parent=2044"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/categories?post=2044"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/tags?post=2044"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}