1.If a theatre holds 1,800 people, and sells ( 1)\/(3) of its seats, how many tickets does it sell?<\/p>\n\n\n\n
2.Find the vector parametrization for the line segment that begins<\/strong><\/p>\n\n\n\n at P ( 2, 7, -1) and ends at Q ( 4, 2, 3 ).<\/strong><\/p>\n\n\n\n The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n The theater holds 1,800 people. Since it sells 13\\frac{1}{3} of its seats, the number of tickets sold is: 13\u00d71800=600\\frac{1}{3} \\times 1800 = 600<\/p>\n\n\n\n So, the theater sells 600 tickets<\/strong>.<\/p>\n\n\n\n To parametrize the line segment from P(2, 7, -1)<\/strong> to Q(4, 2, 3)<\/strong>, we use the formula: r(t)=P+t(Q\u2212P),0\u2264t\u22641\\mathbf{r}(t) = \\mathbf{P} + t (\\mathbf{Q} – \\mathbf{P}), \\quad 0 \\leq t \\leq 1<\/p>\n\n\n\n The direction vector PQ<\/strong> is: PQ=Q\u2212P=(4\u22122,2\u22127,3\u2212(\u22121))=(2,\u22125,4)\\mathbf{PQ} = Q – P = (4 – 2, 2 – 7, 3 – (-1)) = (2, -5, 4)<\/p>\n\n\n\n Since a point on the line is given by: r(t)=(2,7,\u22121)+t(2,\u22125,4),0\u2264t\u22641\\mathbf{r}(t) = (2, 7, -1) + t(2, -5, 4), \\quad 0 \\leq t \\leq 1<\/p>\n\n\n\n Breaking it into components: x=2+2tx = 2 + 2t y=7\u22125ty = 7 – 5t z=\u22121+4tz = -1 + 4t<\/p>\n\n\n\n where 0\u2264t\u226410 \\leq t \\leq 1.<\/p>\n\n\n\n Thus, the vector parametrization<\/strong> of the line segment is: (x,y,z)=(2+2t,7\u22125t,\u22121+4t),0\u2264t\u22641( x, y, z ) = ( 2 + 2t, 7 – 5t, -1 + 4t ), \\quad 0 \\leq t \\leq 1<\/p>\n\n\n\n A vector parametrization is a way of representing a line segment using a parameter tt, which varies within a specific interval. The line segment from point P(2,7,-1)<\/strong> to Q(4,2,3)<\/strong> can be defined using vector operations.<\/p>\n\n\n\n The general equation for a line passing through two points P<\/strong> and Q<\/strong> in three-dimensional space is: r(t)=P+t(Q\u2212P)\\mathbf{r}(t) = \\mathbf{P} + t(\\mathbf{Q} – \\mathbf{P})<\/p>\n\n\n\n where tt is a real number that determines the position on the line.<\/p>\n\n\n\n To construct the equation, we first determine the direction vector PQ<\/strong> by subtracting the coordinates of P from Q: (4\u22122,2\u22127,3\u2212(\u22121))=(2,\u22125,4)(4-2, 2-7, 3 – (-1)) = (2, -5, 4)<\/p>\n\n\n\n This vector gives the direction in which the line moves. Using this, we write the parametric equations: x=2+2t,y=7\u22125t,z=\u22121+4t,0\u2264t\u22641x = 2 + 2t, \\quad y = 7 – 5t, \\quad z = -1 + 4t, \\quad 0 \\leq t \\leq 1<\/p>\n\n\n\n The values of tt range from 0 to 1 because at t=0t = 0, we are at P(2,7,-1), and at t=1t = 1, we reach Q(4,2,3). As tt varies between 0 and 1, the point moves along the line segment.<\/p>\n\n\n\n This form of parametrization is useful in physics, computer graphics, and engineering, as it allows easy representation of motion, interpolation, and geometric calculations.<\/p>\n\n\n\n Now, I’ll generate an image to visualize the line segment.<\/p>\n\n\n\n Here is the visualization of the line segment from P(2,7,-1)<\/strong> to Q(4,2,3)<\/strong> in 3D space. The red dot represents point P, the green dot represents point Q, and the blue line shows the line segment connecting them. Let me know if you need any modifications!<\/p>\n","protected":false},"excerpt":{"rendered":" 1.If a theatre holds 1,800 people, and sells ( 1)\/(3) of its seats, how many tickets does it sell? 2.Find the vector parametrization for the line segment that begins at P ( 2, 7, -1) and ends at Q ( 4, 2, 3 ). The correct answer and explanation is: Solution 1: Number of Tickets […]<\/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-196645","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196645","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=196645"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196645\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196645"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196645"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196645"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Solution 1: Number of Tickets Sold<\/h3>\n\n\n\n
\n\n\n\nSolution 2: Vector Parametrization of a Line Segment<\/h3>\n\n\n\n
Step 1: Find the Direction Vector<\/h4>\n\n\n\n
Step 2: Write the Parametric Equations<\/h4>\n\n\n\n
\n\n\n\nExplanation<\/h3>\n\n\n\n