The resultant of two vectors of magnitude 3A and a

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T17:20:18+00:00","modifiedTime":"2016-03-26T17:20:18+00:00","timestamp":"2022-06-22T19:25:01+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.dummies.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"How to Find the Angle and Magnitude of a Vector","strippedTitle":"how to find the angle and magnitude of a vector","slug":"how-to-find-the-angle-and-magnitude-of-a-vector","canonicalUrl":"","seo":{"metaDescription":"In physics, sometimes you have to find the angle and magnitude of a vector rather than the components. To find the magnitude, you use the Pythagorean theorem. A","noIndex":0,"noFollow":0},"content":"<p>In physics, sometimes you have to find the angle and magnitude of a vector rather than the components. To find the magnitude, you use the Pythagorean theorem. And to find </p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyMy5pbWFnZTAucG5n.webp\" width=\"15\" height=\"19\" alt=\"image0.png\"/>\n<p>you use the inverse tangent function (or inverse sine or cosine).</p>\n<p>For example, assume you’re looking for a hotel that’s 20 miles due east and then 20 miles due north. From your present location, what is the angle (measured from east) of the direction to the hotel, and how far away is the hotel? You can write this problem in vector notation, like so:</p>\n<p>Step 1: (20, 0)</p>\n<p>Step 2: (0, 20)</p>\n<p>When adding these vectors together, you get this result:</p>\n<p> (20, 0) + (0, 20) = (20, 20)</p>\n<p>The resultant vector is (20, 20). That’s one way of specifying a vector — use its components. But this problem isn’t asking for the results in terms of components. The question wants to know the angle and distance to the hotel. </p>\n<div class=\"imageBlock\" style=\"width:386px;\"><img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyNC5pbWFnZTEuanBn.webp\" width=\"386\" height=\"400\" alt=\"Using the angle created by a vector to get to a hotel.\"/><div class=\"imageCaption\">Using the angle created by a vector to get to a hotel.</div></div>\n<p>In other words, looking at the above figure, the problem asks, </p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyNS5pbWFnZTIucG5n.webp\" width=\"165\" height=\"19\" alt=\"image2.png\"/>\n<p>If you know a vector’s vertical and horizontal components, finding the vector’s magnitude isn’t so hard because you just need to find the hypotenuse of a right triangle. You can use the Pythagorean theorem (<i>x</i>² + <i>y</i>² = <i>h</i>²), solved for <i>h</i>:</p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyNi5pbWFnZTMucG5n.webp\" width=\"89\" height=\"31\" alt=\"image3.png\"/>\n<p>Plugging in the numbers gives you</p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyNy5pbWFnZTQucG5n.webp\" width=\"160\" height=\"171\" alt=\"image4.png\"/>\n<p>Keep in mind that when you know the horizontal and vertical components of a vector, you can use the tangent to find the angle because </p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyOC5pbWFnZTUucG5n.webp\" width=\"71\" height=\"40\" alt=\"image5.png\"/>\n<p>All you have to do is take the inverse tangent of <i>y</i>/<i>x</i>:</p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMyOS5pbWFnZTYucG5n.webp\" width=\"92\" height=\"45\" alt=\"image6.png\"/>\n<p>Suppose you drive 20 miles east and 20 miles north. Here’s how you find </p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMzMC5pbWFnZTcucG5n.webp\" width=\"15\" height=\"19\" alt=\"image7.png\"/>\n<p>the angle between your original position and your final one:</p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMzMS5pbWFnZTgucG5n.webp\" width=\"116\" height=\"155\" alt=\"image8.png\"/>\n<p>So the hotel is about 28 miles away at an angle of 45 degrees.</p>\n<p class=\"Warning\">Be careful when doing calculations with inverse tangents, because angles that differ by 180 degrees have the same tangent. When you take the inverse tangent, you may need to add or subtract 180 degrees to get the actual angle you want. The inverse tangent button on your calculator will always give you an angle between 90 degrees and –90 degrees. If your angle is not in this range, then you have to add or subtract 180 degrees.</p>\n<p>For this example, the answer of 45 degrees must be correct. But consider a situation in which you’d need to add or subtract 180 degrees: Suppose that you walk in completely the opposite direction to the hotel. You walk 20 miles west and 20 miles south (<i>x</i> = –20 miles, <i>y</i> = –20 miles), so if you use the same method to work out the angle, you get the following:</p>\n<img src=\"//sg.cdnki.com/the-resultant-of-two-vectors-of-magnitude-3a-and-a---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzMzMjMzMi5pbWFnZTkucG5n.webp\" width=\"124\" height=\"155\" alt=\"image9.png\"/>\n<p>You get the same answer for the angle even though you’re walking in completely the opposite direction as before! That’s because the tangents of angles that differ by 180 degrees are equal. But if you look at the components of the vector (<i>x</i> = –20 miles, <i>y</i> = –20 miles), they’re both negative, so the angle must be between -90 degrees and -180 degrees. If you subtract 180 degrees from your answer of 45 degrees, you get –135 degrees, which is your actual angle measured from the positive x-axis in the clockwise direction. </p>\n<p>Alternatively, you could reason that since the components of the vector are both negative, you must be between 180 degrees and 270 degrees. You would then add 180 degrees to your result and get 225 degrees, which would be measured from the positive x-axis in the counterclockwise direction. </p>\n<p>So, which angle is correct, 225 degrees or -135 degrees? Both! Whether you move counterclockwise 225 degrees or clockwise 135 degrees from the positive x-axis, you end up heading the same direction.</p>","description":"<p>In physics, sometimes you have to find the angle and magnitude of a vector rather than the components. To find the magnitude, you use the Pythagorean theorem. And to find </p>\n<img src=\"//www.dummies.com/wp-content/uploads/332323.image0.png\" width=\"15\" height=\"19\" alt=\"image0.png\"/>\n<p>you use the inverse tangent function (or inverse sine or cosine).</p>\n<p>For example, assume you’re looking for a hotel that’s 20 miles due east and then 20 miles due north. From your present location, what is the angle (measured from east) of the direction to the hotel, and how far away is the hotel? You can write this problem in vector notation, like so:</p>\n<p>Step 1: (20, 0)</p>\n<p>Step 2: (0, 20)</p>\n<p>When adding these vectors together, you get this result:</p>\n<p> (20, 0) + (0, 20) = (20, 20)</p>\n<p>The resultant vector is (20, 20). That’s one way of specifying a vector — use its components. But this problem isn’t asking for the results in terms of components. The question wants to know the angle and distance to the hotel. </p>\n<div class=\"imageBlock\" style=\"width:386px;\"><img src=\"//www.dummies.com/wp-content/uploads/332324.image1.jpg\" width=\"386\" height=\"400\" alt=\"Using the angle created by a vector to get to a hotel.\"/><div class=\"imageCaption\">Using the angle created by a vector to get to a hotel.</div></div>\n<p>In other words, looking at the above figure, the problem asks, </p>\n<img src=\"//www.dummies.com/wp-content/uploads/332325.image2.png\" width=\"165\" height=\"19\" alt=\"image2.png\"/>\n<p>If you know a vector’s vertical and horizontal components, finding the vector’s magnitude isn’t so hard because you just need to find the hypotenuse of a right triangle. You can use the Pythagorean theorem (<i>x</i>² + <i>y</i>² = <i>h</i>²), solved for <i>h</i>:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/332326.image3.png\" width=\"89\" height=\"31\" alt=\"image3.png\"/>\n<p>Plugging in the numbers gives you</p>\n<img src=\"//www.dummies.com/wp-content/uploads/332327.image4.png\" width=\"160\" height=\"171\" alt=\"image4.png\"/>\n<p>Keep in mind that when you know the horizontal and vertical components of a vector, you can use the tangent to find the angle because </p>\n<img src=\"//www.dummies.com/wp-content/uploads/332328.image5.png\" width=\"71\" height=\"40\" alt=\"image5.png\"/>\n<p>All you have to do is take the inverse tangent of <i>y</i>/<i>x</i>:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/332329.image6.png\" width=\"92\" height=\"45\" alt=\"image6.png\"/>\n<p>Suppose you drive 20 miles east and 20 miles north. Here’s how you find </p>\n<img src=\"//www.dummies.com/wp-content/uploads/332330.image7.png\" width=\"15\" height=\"19\" alt=\"image7.png\"/>\n<p>the angle between your original position and your final one:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/332331.image8.png\" width=\"116\" height=\"155\" alt=\"image8.png\"/>\n<p>So the hotel is about 28 miles away at an angle of 45 degrees.</p>\n<p class=\"Warning\">Be careful when doing calculations with inverse tangents, because angles that differ by 180 degrees have the same tangent. When you take the inverse tangent, you may need to add or subtract 180 degrees to get the actual angle you want. The inverse tangent button on your calculator will always give you an angle between 90 degrees and –90 degrees. If your angle is not in this range, then you have to add or subtract 180 degrees.</p>\n<p>For this example, the answer of 45 degrees must be correct. But consider a situation in which you’d need to add or subtract 180 degrees: Suppose that you walk in completely the opposite direction to the hotel. You walk 20 miles west and 20 miles south (<i>x</i> = –20 miles, <i>y</i> = –20 miles), so if you use the same method to work out the angle, you get the following:</p>\n<img src=\"//www.dummies.com/wp-content/uploads/332332.image9.png\" width=\"124\" height=\"155\" alt=\"image9.png\"/>\n<p>You get the same answer for the angle even though you’re walking in completely the opposite direction as before! That’s because the tangents of angles that differ by 180 degrees are equal. But if you look at the components of the vector (<i>x</i> = –20 miles, <i>y</i> = –20 miles), they’re both negative, so the angle must be between -90 degrees and -180 degrees. If you subtract 180 degrees from your answer of 45 degrees, you get –135 degrees, which is your actual angle measured from the positive x-axis in the clockwise direction. </p>\n<p>Alternatively, you could reason that since the components of the vector are both negative, you must be between 180 degrees and 270 degrees. You would then add 180 degrees to your result and get 225 degrees, which would be measured from the positive x-axis in the counterclockwise direction. </p>\n<p>So, which angle is correct, 225 degrees or -135 degrees? Both! Whether you move counterclockwise 225 degrees or clockwise 135 degrees from the positive x-axis, you end up heading the same direction.</p>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","_links":{"self":"//dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.dummies.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/208460"}},{"articleId":184049,"title":"A List of Physics Constants","slug":"a-list-of-physics-constants","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/184049"}},{"articleId":184043,"title":"Physics Equations and Formulas","slug":"physics-equations-and-formulas","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/184043"}},{"articleId":174308,"title":"Calculating Tangential Velocity on a Curve","slug":"calculating-tangential-velocity-on-a-curve","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/174308"}},{"articleId":174307,"title":"Flowing from Hot to Cold: The Second Law of Thermodynamics","slug":"flowing-from-hot-to-cold-the-second-law-of-thermodynamics","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/174307"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282467,"slug":"physics-i-for-dummies","isbn":"9781119872221","categoryList":["academics-the-arts","science","physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119872227-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//www.dummies.com/wp-content/uploads/9781119872221-203x255.jpg","width":203,"height":255},"title":"Physics I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b> Dr. <b data-author-id=\"8967\">Steven Holzner</b></b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","_links":{"self":"//dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"//dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-62b36c8d40050\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-62b36c8d40b34\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":null,"sponsorEbookTitle":null,"sponsorEbookLink":null,"sponsorEbookImage":null},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":173966},"articleLoadedStatus":"success"},"listState":{"list":{},"objectTitle":"","status":"initial","pageType":null,"objectId":null,"page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{},"filterDataLoadedStatus":"initial","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2022-07-22T12:59:03+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely Script-->\r\n<script src=\"//cdn.optimizely.com/js/10563184655.js\"></script>","enabled":false},{"pages":["all"],"location":"header","script":"<!-- comScore Tag -->\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \"2\", c2: \"15097263\" });(function() {var s = document.createElement(\"script\"), el = document.getElementsByTagName(\"script\")[0]; s.async = true;s.src = (document.location.protocol == \"\" ? \"//sb\" : \"//b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();</script><noscript><img src=\"//sb.scorecardresearch.com/p?c1=2&c2=15097263&cv=2.0&cj=1\" /></noscript>\r\n<!-- / comScore Tag -->","enabled":true},{"pages":["all"],"location":"footer","script":"<!--BEGIN QUALTRICS WEBSITE FEEDBACK SNIPPET-->\r\n<script type='text/javascript'>\r\n(function(){var g=function(e,h,f,g){\r\nthis.get=function(a){for(var a=a+\"=\",c=document.cookie.split(\";\"),b=0,e=c.length;b<e;b++){for(var d=c[b];\" \"==d.charAt(0);)d=d.substring(1,d.length);if(0==d.indexOf(a))return d.substring(a.length,d.length)}return null};\r\nthis.set=function(a,c){var b=\"\",b=new Date;b.setTime(b.getTime()+6048E5);b=\"; expires=\"+b.toGMTString();document.cookie=a+\"=\"+c+b+\"; path=/; \"};\r\nthis.check=function(){var a=this.get(f);if(a)a=a.split(\":\");else if(100!=e)\"v\"==h&&(e=Math.random()>=e/100?0:100),a=[h,e,0],this.set(f,a.join(\":\"));else return!0;var c=a[1];if(100==c)return!0;switch(a[0]){case \"v\":return!1;case \"r\":return c=a[2]%Math.floor(100/c),a[2]++,this.set(f,a.join(\":\")),!c}return!0};\r\nthis.go=function(){if(this.check()){var a=document.createElement(\"script\");a.type=\"text/javascript\";a.src=g;document.body&&document.body.appendChild(a)}};\r\nthis.start=function(){var t=this;\"complete\"!==document.readyState?window.addEventListener?window.addEventListener(\"load\",function(){t.go()},!1):window.attachEvent&&window.attachEvent(\"onload\",function(){t.go()}):t.go()};};\r\ntry{(new g(100,\"r\",\"QSI_S_ZN_5o5yqpvMVjgDOuN\",\"//zn5o5yqpvmvjgdoun-wiley.siteintercept.qualtrics.com/SIE/?Q_ZID=ZN_5o5yqpvMVjgDOuN\")).start()}catch(i){}})();\r\n</script><div id='ZN_5o5yqpvMVjgDOuN'><!--DO NOT REMOVE-CONTENTS PLACED HERE--></div>\r\n<!--END WEBSITE FEEDBACK SNIPPET-->","enabled":false},{"pages":["all"],"location":"header","script":"<!-- Hotjar Tracking Code for //www.dummies.com -->\r\n<script>\r\n (function(h,o,t,j,a,r){\r\n h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)};\r\n h._hjSettings={hjid:257151,hjsv:6};\r\n a=o.getElementsByTagName('head')[0];\r\n r=o.createElement('script');r.async=1;\r\n r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv;\r\n a.appendChild(r);\r\n })(window,document,'//static.hotjar.com/c/hotjar-','.js?sv=');\r\n</script>","enabled":false},{"pages":["article"],"location":"header","script":"<!-- //Connect Container: dummies --> <script src=\"//get.s-onetag.com/bffe21a1-6bb8-4928-9449-7beadb468dae/tag.min.js\" async defer></script>","enabled":true},{"pages":["homepage"],"location":"header","script":"<meta name=\"facebook-domain-verification\" content=\"irk8y0irxf718trg3uwwuexg6xpva0\" />","enabled":true},{"pages":["homepage","article","category","search"],"location":"footer","script":"<!-- Facebook Pixel Code -->\r\n<noscript>\r\n<img height=\"1\" width=\"1\" src=\"//www.facebook.com/tr?id=256338321977984&ev=PageView&noscript=1\"/>\r\n</noscript>\r\n<!-- End Facebook Pixel Code -->","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":287569,"title":"For the Hopeless Romantic","hasSubCategories":false,"url":"/collection/for-the-hopeless-romantic-287569"},{"collectionId":287567,"title":"For the Unabashed Hippie","hasSubCategories":false,"url":"/collection/for-the-unabashed-hippie-287567"},{"collectionId":292186,"title":"Just DIY It","hasSubCategories":false,"url":"/collection/just-diy-it-292186"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"Article3","path":"/article/academics-the-arts/science/physics/how-to-find-the-angle-and-magnitude-of-a-vector-173966/","hash":"","query":{},"params":{"category1":"academics-the-arts","category2":"science","category3":"physics","article":"how-to-find-the-angle-and-magnitude-of-a-vector-173966"},"fullPath":"/article/academics-the-arts/science/physics/how-to-find-the-angle-and-magnitude-of-a-vector-173966/","meta":{"routeType":"article","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"dropsState":{"submitEmailResponse":false,"status":"initial"},"sfmcState":{"newsletterSignupStatus":"initial"}}