Consider the vector v with rectangular components (a, b) and polar components (r, \u03b8).<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The horizontal component ( a ) can be expressed in terms of the polar components as follows:<\/p>\n\n\n\n [ Explanation:<\/strong> The vertical component ( b ) is given by:<\/p>\n\n\n\n [ Explanation:<\/strong> The magnitude ( r ) is related to ( a ) and ( b ) by the Pythagorean theorem:<\/p>\n\n\n\n [ Explanation:<\/strong> The angle ( \\theta ) is given by:<\/p>\n\n\n\n [ Explanation:<\/strong> Consider the vector v with rectangular components (a, b) and polar components (r, \u03b8). The Correct Answer and Explanation is : 1. Relationship between ( r ) and the horizontal component ( a ): The horizontal component ( a ) can be expressed in terms of the polar components as follows: [a = r \\cos(\\theta)] […]<\/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-189772","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189772","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=189772"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189772\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=189772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=189772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=189772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}1. Relationship between ( r ) and the horizontal component ( a ):<\/h3>\n\n\n\n
a = r \\cos(\\theta)
]<\/p>\n\n\n\n
In polar coordinates, ( r ) represents the magnitude (or length) of the vector, and ( \\theta ) represents the angle the vector makes with the positive x-axis. The horizontal component of the vector, ( a ), is found by multiplying the magnitude ( r ) by the cosine of the angle ( \\theta ). This uses basic trigonometry, as the cosine function gives the adjacent side of a right triangle in a unit circle.<\/p>\n\n\n\n2. Relationship between ( r ) and the vertical component ( b ):<\/h3>\n\n\n\n
b = r \\sin(\\theta)
]<\/p>\n\n\n\n
Similarly to the horizontal component, the vertical component is the projection of the vector onto the y-axis. The sine function gives the ratio of the opposite side in a right triangle, so multiplying the magnitude ( r ) by ( \\sin(\\theta) ) gives the length of the vertical component ( b ). This is a direct application of trigonometry.<\/p>\n\n\n\n3. Relationship between ( a ), ( b ), and the magnitude ( r ):<\/h3>\n\n\n\n
r = \\sqrt{a^2 + b^2}
]<\/p>\n\n\n\n
The magnitude of the vector is the hypotenuse of a right triangle whose sides are the components ( a ) (horizontal) and ( b ) (vertical). By the Pythagorean theorem, the magnitude ( r ) is the square root of the sum of the squares of ( a ) and ( b ).<\/p>\n\n\n\n4. Relationship between ( a ), ( b ), and the angle ( \\theta ):<\/h3>\n\n\n\n
\\theta = \\tan^{-1}\\left(\\frac{b}{a}\\right)
]<\/p>\n\n\n\n
The angle ( \\theta ) can be found using the arctangent function, which is the inverse of the tangent. The tangent of an angle in a right triangle is the ratio of the opposite side (vertical component ( b )) to the adjacent side (horizontal component ( a )). Therefore, ( \\theta ) can be calculated as the arctangent of ( b\/a ), which gives the angle the vector makes with the positive x-axis.<\/p>\n","protected":false},"excerpt":{"rendered":"