{"id":187322,"date":"2025-02-04T06:34:02","date_gmt":"2025-02-04T06:34:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=187322"},"modified":"2025-02-04T06:34:04","modified_gmt":"2025-02-04T06:34:04","slug":"find-the-volume-of-the-parallelepiped-determined-by-the-vectors-a-i-j-b-j-k-and-cijk-6","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/04\/find-the-volume-of-the-parallelepiped-determined-by-the-vectors-a-i-j-b-j-k-and-cijk-6\/","title":{"rendered":"Find the volume of the parallelepiped determined by the vectors a = i + j, b = j + k, and c=i+j+k 6"},"content":{"rendered":"\n<p>. Find the volume of the parallelepiped determined by the vectors a = i + j, b = j + k, and c=i+j+k 6. Given that P(3,0,1),Q(\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221,2,5),R(5,1,\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221), and S(0,4,2). Find the volume of the paral- lelepiped with adjacent edges PQ,PR, and PS. 7. Given that P (1, 1, 1), Q(2, 1, 3), and R(3, \u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221, 1) are vertices of a triangle. (a) Find the area of the triangle determined by the points P, Q, and R. (b) Find a unit vector perpendicular to plane PQR. 8. Leta=i+2j\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u2122k,b=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u2122i+j+k,andc=i+k.Whichvectors,ifany,are (a) perpendicular? (b) Parallel? Give reasons for your answers. 5. Find the volume of the parallelepiped determined by the vectors a = i + j, b = j + k, and c=i+j+k 6. Given that P(3,0,1),Q(\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221,2,5),R(5,1,\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221), and S(0,4,2). Find the volume of the paral- lelepiped with adjacent edges PQ,PR, and PS. 7. Given that P (1, 1, 1), Q(2, 1, 3), and R(3, \u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221, 1) are vertices of a triangle. (a) Find the area of the triangle determined by the points P, Q, and R. (b) Find a unit vector perpendicular to plane PQR. 8. Leta=i+2j\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u2122k,b=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u2122i+j+k,andc=i+k.Whichvectors,ifany,are (a) perpendicular? (b) Parallel? Give reasons for your answers.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s address each problem step by step:<\/p>\n\n\n\n<p><strong>5. Volume of the Parallelepiped Determined by Vectors a, b, and c<\/strong><\/p>\n\n\n\n<p>Given vectors:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>a<\/strong> = i + j = (1, 1, 0)<\/li>\n\n\n\n<li><strong>b<\/strong> = j + k = (0, 1, 1)<\/li>\n\n\n\n<li><strong>c<\/strong> = i + j + k = (1, 1, 1)<\/li>\n<\/ul>\n\n\n\n<p>The volume ( V ) of a parallelepiped formed by vectors <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong> is given by the absolute value of the scalar triple product:<\/p>\n\n\n\n<p>[ V = |<strong>a<\/strong> \\cdot (<strong>b<\/strong> \\times <strong>c<\/strong>)| ]<\/p>\n\n\n\n<p>First, compute the cross product <strong>b<\/strong> \u00d7 <strong>c<\/strong>:<\/p>\n\n\n\n<p>[<br>\\begin{vmatrix}<br>\\mathbf{i} &amp; \\mathbf{j} &amp; \\mathbf{k} \\<br>0 &amp; 1 &amp; 1 \\<br>1 &amp; 1 &amp; 1<br>\\end{vmatrix}<br>= \\mathbf{i}(1 \\cdot 1 &#8211; 1 \\cdot 1) &#8211; \\mathbf{j}(0 \\cdot 1 &#8211; 1 \\cdot 1) + \\mathbf{k}(0 \\cdot 1 &#8211; 1 \\cdot 1)<br>= \\mathbf{i}(0) &#8211; \\mathbf{j}(-1) + \\mathbf{k}(-1)<br>= \\mathbf{j} &#8211; \\mathbf{k}<br>]<\/p>\n\n\n\n<p>So, <strong>b<\/strong> \u00d7 <strong>c<\/strong> = j &#8211; k = (0, 1, -1).<\/p>\n\n\n\n<p>Next, compute the dot product <strong>a<\/strong> \u00b7 (<strong>b<\/strong> \u00d7 <strong>c<\/strong>):<\/p>\n\n\n\n<p>[<br>(1, 1, 0) \\cdot (0, 1, -1) = 1 \\cdot 0 + 1 \\cdot 1 + 0 \\cdot (-1) = 0 + 1 + 0 = 1<br>]<\/p>\n\n\n\n<p>Therefore, the volume ( V ) is:<\/p>\n\n\n\n<p>[ V = |1| = 1 ]<\/p>\n\n\n\n<p>The volume of the parallelepiped is 1 cubic unit.<\/p>\n\n\n\n<p><strong>6. Volume of the Parallelepiped with Adjacent Edges PQ, PR, and PS<\/strong><\/p>\n\n\n\n<p>Given points:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P(3, 0, 1)<\/li>\n\n\n\n<li>Q(-1, 2, 5)<\/li>\n\n\n\n<li>R(5, 1, -1)<\/li>\n\n\n\n<li>S(0, 4, 2)<\/li>\n<\/ul>\n\n\n\n<p>First, determine vectors PQ, PR, and PS:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>PQ<\/strong> = Q &#8211; P = (-1 &#8211; 3, 2 &#8211; 0, 5 &#8211; 1) = (-4, 2, 4)<\/li>\n\n\n\n<li><strong>PR<\/strong> = R &#8211; P = (5 &#8211; 3, 1 &#8211; 0, -1 &#8211; 1) = (2, 1, -2)<\/li>\n\n\n\n<li><strong>PS<\/strong> = S &#8211; P = (0 &#8211; 3, 4 &#8211; 0, 2 &#8211; 1) = (-3, 4, 1)<\/li>\n<\/ul>\n\n\n\n<p>The volume ( V ) is given by the absolute value of the scalar triple product:<\/p>\n\n\n\n<p>[ V = |<strong>PQ<\/strong> \\cdot (<strong>PR<\/strong> \\times <strong>PS<\/strong>)| ]<\/p>\n\n\n\n<p>Compute the cross product <strong>PR<\/strong> \u00d7 <strong>PS<\/strong>:<\/p>\n\n\n\n<p>[<br>\\begin{vmatrix}<br>\\mathbf{i} &amp; \\mathbf{j} &amp; \\mathbf{k} \\<br>2 &amp; 1 &amp; -2 \\<br>-3 &amp; 4 &amp; 1<br>\\end{vmatrix}<br>= \\mathbf{i}(1 \\cdot 1 &#8211; (-2) \\cdot 4) &#8211; \\mathbf{j}(2 \\cdot 1 &#8211; (-2) \\cdot (-3)) + \\mathbf{k}(2 \\cdot 4 &#8211; 1 \\cdot (-3))<br>= \\mathbf{i}(1 + 8) &#8211; \\mathbf{j}(2 &#8211; 6) + \\mathbf{k}(8 + 3)<br>= 9\\mathbf{i} + 4\\mathbf{j} + 11\\mathbf{k}<br>]<\/p>\n\n\n\n<p>So, <strong>PR<\/strong> \u00d7 <strong>PS<\/strong> = (9, 4, 11).<\/p>\n\n\n\n<p>Next, compute the dot product <strong>PQ<\/strong> \u00b7 (<strong>PR<\/strong> \u00d7 <strong>PS<\/strong>):<\/p>\n\n\n\n<p>[<br>(-4, 2, 4) \\cdot (9, 4, 11) = (-4) \\cdot 9 + 2 \\cdot 4 + 4 \\cdot 11 = -36 + 8 + 44 = 16<br>]<\/p>\n\n\n\n<p>Therefore, the volume ( V ) is:<\/p>\n\n\n\n<p>[ V = |16| = 16 ]<\/p>\n\n\n\n<p>The volume of the parallelepiped is 16 cubic units.<\/p>\n\n\n\n<p><strong>7. Triangle with Vertices P, Q, and R<\/strong><\/p>\n\n\n\n<p>Given points:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P(1, 1, 1)<\/li>\n\n\n\n<li>Q(2, 1, 3)<\/li>\n\n\n\n<li>R(3, -1, 1)<\/li>\n<\/ul>\n\n\n\n<p>(a) <strong>Area of the Triangle<\/strong><\/p>\n\n\n\n<p>First, determine vectors PQ and PR:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>PQ<\/strong> = Q &#8211; P = (2 &#8211; 1, 1 &#8211; 1, 3 &#8211; 1) = (1, 0, 2)<\/li>\n\n\n\n<li><strong>PR<\/strong> = R &#8211; P = (3 &#8211; 1, -1 &#8211; 1, 1 &#8211; 1) = (2, -2, 0)<\/li>\n<\/ul>\n\n\n\n<p>Compute the cross product <strong>PQ<\/strong> \u00d7 <strong>PR<\/strong>:<\/p>\n\n\n\n<p>[<br>\\begin{vmatrix}<br>\\mathbf{i} &amp; \\mathbf{j} &amp; \\mathbf{k} \\<br>1 &amp; 0 &amp; 2 \\<br>2 &amp; -2 &amp; 0<br>\\end{vmatrix}<br>= \\mathbf{i}(0 \\cdot 0 &#8211; 2 \\cdot (-2)) &#8211; \\mathbf{j}(1 \\cdot 0 &#8211; 2 \\<\/p>\n","protected":false},"excerpt":{"rendered":"<p>. Find the volume of the parallelepiped determined by the vectors a = i + j, b = j + k, and c=i+j+k 6. Given that P(3,0,1),Q(\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221,2,5),R(5,1,\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221), and S(0,4,2). Find the volume of the paral- lelepiped with adjacent edges PQ,PR, and PS. 7. Given that P (1, 1, 1), Q(2, 1, 3), and R(3, \u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21221, [&hellip;]<\/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-187322","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187322","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=187322"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187322\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187322"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187322"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187322"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}