{"id":25505120,"date":"2022-01-21T13:30:13","date_gmt":"2022-01-21T08:00:13","guid":{"rendered":"https:\/\/entri.app\/blog\/?p=25505120"},"modified":"2022-01-21T13:30:13","modified_gmt":"2022-01-21T08:00:13","slug":"basics-of-vectors","status":"publish","type":"post","link":"https:\/\/entri.app\/blog\/basics-of-vectors\/","title":{"rendered":"Basics of Vectors &#8211; Mathematics Notes for Competitive Exam"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_2 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69e7516a29c38\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69e7516a29c38\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/entri.app\/blog\/basics-of-vectors\/#Basics_of_vectors_Notes\" >Basics of vectors Notes<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/entri.app\/blog\/basics-of-vectors\/#Meaning_of_Position_Vector\" >Meaning of Position Vector<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/entri.app\/blog\/basics-of-vectors\/#Direction_Cosines\" >Direction Cosines<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>In this article, we will discuss the Basics of Vectors which is one of the very important topics that need to be covered in the preparation for competitive exams. Here we shall cover the basics and definition of vectors. We will also have a look at the meaning of a vector by understanding the basic components\u00a0of a vector.<\/p>\n<p style=\"text-align: center\"><strong><a href=\"https:\/\/entri.sng.link\/Bcofz\/od54\/6blp\" target=\"_blank\" rel=\"noopener\">Prepare for various Government exams! Download Entri App!<\/a><\/strong><\/p>\n<h1><span class=\"ez-toc-section\" id=\"Basics_of_vectors_Notes\"><\/span><strong>Basics of vectors Notes<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>While studying math and physics, we come across two types of quantities namely scalars and vectors. The quantity that consists of only value or magnitude but no direction is a Scalar quantity, for example, length, mass, speed, area, etc. On the other hand, the quantity that involves magnitude, as well as direction, is a Vector quantity, for example, displacement, velocity, acceleration, force, weight, momentum, etc.<\/p>\n<p>Basic Definition \u00a0a Vector<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-25505130 aligncenter\" src=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture1-1-300x166.png\" alt=\"\" width=\"427\" height=\"235\" srcset=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture1-1-300x166.png 300w, https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture1-1.png 335w\" sizes=\"auto, (max-width: 427px) 100vw, 427px\" \/><\/p>\n<p style=\"text-align: center\"><strong>Fig. 1<\/strong><\/p>\n<ul>\n<li>A measure that has magnitude, as well as direction, is called a vector. From the figure above, the directed line segment shown in Figure 1 (iii) is a vector.<\/li>\n<li>It is represented as \u2018vector\u00a0\u00a0AB\u2192\u2019 or \u2018vector a\u2192 \u2019.<\/li>\n<li>point A from where the vector AB\u2192 begins is called its starting or initial point, and point B where it terminates is called its end or terminal point.<\/li>\n<li>The\u00a0distance between these two points of a vector namely the initial and terminal is called the magnitude or length of the vector, denoted as |a\u2192 |, or |AB\u2192| or The arrow indicates the vector\u2019s direction.<\/li>\n<\/ul>\n<p>Note: Length is always positive or never negative. Hence,\u00a0AB\u2192\u00a0| is always\u00a0\u2265 0.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Meaning_of_Position_Vector\"><\/span><strong><b>Meaning of Position Vector<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Have a\u00a0look at the\u00a0following diagram\u00a0below<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-25505131 aligncenter\" src=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture2.png\" alt=\"\" width=\"352\" height=\"299\" \/><\/p>\n<p style=\"text-align: center\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<strong>Fig.2<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>Figure\u00a0 2 shows a right-handed rectangular or Cartesian coordinate system.<\/li>\n<li>Let\u2019s consider a point \u2018P\u2019 in space, such that its coordinates are (x, y, z) with respect to the origin O (0, 0, 0).<\/li>\n<li>The position vector of the p with respect to the O is the vector OP\u2192, where O is the starting point and P is the end point of the vector.<\/li>\n<li>As we already know about the distance formula. We will apply it to find the magnitude of OP\u2192 or r\u2192 , here, we get<\/li>\n<\/ul>\n<p>|OP\u2192| =\u00a0x2+y2+z2\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u221a<\/p>\n<p>In practice, the position vectors of points A, B, C, \u2026 with respect to the origin O are denoted by a\u2192 , b\u2192 , c\u2192 , \u2026. as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-25505132 aligncenter\" src=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture3.png\" alt=\"\" width=\"416\" height=\"404\" srcset=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture3.png 235w, https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture3-24x24.png 24w, https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture3-48x48.png 48w\" sizes=\"auto, (max-width: 416px) 100vw, 416px\" \/><\/p>\n<p style=\"text-align: center\"><strong><a href=\"https:\/\/entri.sng.link\/Bcofz\/od54\/6blp\" target=\"_blank\" rel=\"noopener\">Prepare for various Government exams! Download Entri App!<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Direction_Cosines\"><\/span><strong><b>Direction Cosines<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Next, have\u00a0a look at the figure below<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-25505133 aligncenter\" src=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture4-300x212.png\" alt=\"\" width=\"446\" height=\"315\" srcset=\"https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture4-300x212.png 300w, https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture4-120x86.png 120w, https:\/\/entri.app\/blog\/wp-content\/uploads\/2022\/01\/Picture4.png 444w\" sizes=\"auto, (max-width: 446px) 100vw, 446px\" \/><\/p>\n<ul>\n<li>Figure 4 shows a position vector OP\u2192 (or r\u2192 ) of a point P (x, y, z). The vector r\u2192\u00a0\u00a0makes angles \u03b1, \u03b2, and \u03b3 with the positive directions of the x, y, and z-axes respectively. These are called direction angles. Now, the direction cosines of the vector r\u2192\u00a0 are the cosine values of these angles. They are denoted as:\n<ul>\n<li>l =\u00a0cos\u03b1<\/li>\n<li>m =\u00a0cos\u03b2<\/li>\n<li>n =\u00a0cos\u03b3<\/li>\n<\/ul>\n<\/li>\n<li>Another thing that you may note from the same, is that the triangle OAP is a right-angled triangle. Hence, we have\n<ul>\n<li>cos\u03b1 = xr, where r stands for |r\u2192 |.<\/li>\n<\/ul>\n<\/li>\n<li>Similarly, from the right-angled triangles OCP and OBP, we have cos\u03b2 = yr and cos\u03b3 = zr<\/li>\n<\/ul>\n<p>Therefore, the coordinates of the point P can be given as (lr, mr, nr). These numbers, lr, mr and nr, are proportional to the direction cosines of the vector r\u2192 . They are called direction ratios of the r\u2192\u00a0\u00a0and are represented as a, b, and c respectively.<\/p>\n<p><a href=\"https:\/\/entri.sng.link\/Bcofz\/od54\/6blp\" target=\"_blank\" rel=\"noopener\"><u>Entri<\/u><\/a>\u00a0app provides you with best practice questions for the complete preparation of various competitive\u00a0exams. For the complete preparation, download the\u00a0<a href=\"https:\/\/www.entri.app\/?referrer=singular_click_id=23102a6b-0576-40a5-bb27-83d7c09edad6\" target=\"_blank\" rel=\"noopener\"><u>Entri<\/u><\/a>\u00a0app and start preparing.<\/p>\n<p>Our R&amp;D team provides the best material and books for the complete preparation of all government exams. We provide PDFs of the previous year&#8217;s question papers with solutions and detailed analysis. All you need to do is download the Entri app and start practicing.<\/p>\n<p style=\"text-align: center\"><strong><a href=\"https:\/\/entri.sng.link\/Bcofz\/od54\/6blp\" target=\"_blank\" rel=\"noopener\">Prepare for various Government exams! Download Entri App!<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this article, we will discuss the Basics of Vectors which is one of the very important topics that need to be covered in the preparation for competitive exams. Here we shall cover the basics and definition of vectors. We will also have a look at the meaning of a vector by understanding the basic [&hellip;]<\/p>\n","protected":false},"author":42,"featured_media":25505136,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[802,1746],"tags":[],"class_list":["post-25505120","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-articles","category-preparation-tips-tricks"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Basics of Vectors - Mathematics Notes for Competitive Exam - Entri Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/entri.app\/blog\/basics-of-vectors\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Basics of Vectors - Mathematics Notes for Competitive Exam - Entri Blog\" \/>\n<meta property=\"og:description\" content=\"In this article, we will discuss the Basics of Vectors which is one of the very important topics that need to be covered in the preparation for competitive exams. 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