{"id":221533,"date":"2025-05-29T18:44:53","date_gmt":"2025-05-29T18:44:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221533"},"modified":"2025-05-29T18:44:56","modified_gmt":"2025-05-29T18:44:56","slug":"wxyz-is-a-parallelogram","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/29\/wxyz-is-a-parallelogram\/","title":{"rendered":"WXYZ is a parallelogram"},"content":{"rendered":"\n

WXYZ is a parallelogram. In the diagram, C is the midpoint of WZ, and A and B trisect XY. Express each vector in terms of a linear combination of r m and . a) XY b)CY c)WX d)WA<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The image you provided shows a parallelogram ABCD<\/strong>, with diagonals intersecting at point E. The red markings indicate that the diagonals bisect each other: AE=ECAE = EC, DE=EBDE = EB, etc. However, your question refers to a different diagram<\/strong> involving parallelogram WXYZ<\/strong>, and points C, A, and B as midpoints or trisection points \u2014 which is not the diagram shown.<\/p>\n\n\n\n

Let’s proceed based on the question’s description<\/strong>, as the image does not align with the question.<\/p>\n\n\n\n

\n\n\n\n

Given:<\/h3>\n\n\n\n
    \n
  • WXYZ is a parallelogram.<\/li>\n\n\n\n
  • C is the midpoint<\/strong> of WZ<\/strong>.<\/li>\n\n\n\n
  • A and B trisect<\/strong> XY<\/strong>, meaning they divide it into three equal segments: XA=AB=BYXA = AB = BY.<\/li>\n\n\n\n
  • Let vectors be defined as:\n
      \n
    • r\u20d7=XY\u20d7\\vec{r} = \\vec{XY}<\/li>\n\n\n\n
    • m\u20d7=WZ\u20d7\\vec{m} = \\vec{WZ}<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

      a) XY\u20d7\\vec{XY}<\/h3>\n\n\n\n

      This is directly given as vector r\u20d7\\vec{r}.<\/p>\n\n\n\n

      Answer<\/strong>:
      XY\u20d7=r\u20d7\\vec{XY} = \\vec{r}<\/p>\n\n\n\n

      \n\n\n\n

      b) CY\u20d7\\vec{CY}<\/h3>\n\n\n\n

      To express CY\u20d7\\vec{CY}, consider the path:<\/p>\n\n\n\n

        \n
      • From C (midpoint of WZ) to W: CW\u20d7=\u221212m\u20d7\\vec{CW} = -\\frac{1}{2} \\vec{m}<\/li>\n\n\n\n
      • Then from W to X: WX\u20d7=\u2212m\u20d7\\vec{WX} = -\\vec{m} (opposite side of parallelogram)<\/li>\n\n\n\n
      • Then from X to Y: XY\u20d7=r\u20d7\\vec{XY} = \\vec{r}<\/li>\n<\/ul>\n\n\n\n

        Thus: CY\u20d7=CW\u20d7+WX\u20d7+XY\u20d7=\u221212m\u20d7\u2212m\u20d7+r\u20d7=\u221232m\u20d7+r\u20d7\\vec{CY} = \\vec{CW} + \\vec{WX} + \\vec{XY} = -\\frac{1}{2} \\vec{m} – \\vec{m} + \\vec{r} = -\\frac{3}{2} \\vec{m} + \\vec{r}<\/p>\n\n\n\n

        Answer<\/strong>:
        CY\u20d7=r\u20d7\u221232m\u20d7\\vec{CY} = \\vec{r} – \\frac{3}{2} \\vec{m}<\/p>\n\n\n\n

        \n\n\n\n

        c) WX\u20d7\\vec{WX}<\/h3>\n\n\n\n

        Since WXYZ is a parallelogram, opposite sides are equal and parallel.<\/p>\n\n\n\n

        Given WZ\u20d7=m\u20d7\\vec{WZ} = \\vec{m}, then:<\/p>\n\n\n\n

          \n
        • WX\u20d7\\vec{WX} is the adjacent side.<\/li>\n<\/ul>\n\n\n\n

          Let\u2019s say W to X is vector a\u20d7\\vec{a}. Since WXYZ is a parallelogram, and XY is r\u20d7\\vec{r}, then WX\u20d7\\vec{WX} plus r\u20d7\\vec{r} equals WY\u20d7\\vec{WY}, a diagonal. But without more detail, we can define:<\/p>\n\n\n\n

          Assume standard parallelogram orientation<\/strong>:<\/p>\n\n\n\n

            \n
          • Let WX\u20d7\\vec{WX} be adjacent to WZ\u20d7\\vec{WZ}, and the diagram aligns XY as the top edge.<\/li>\n<\/ul>\n\n\n\n

            Then:<\/p>\n\n\n\n

            Answer<\/strong>:
            WX\u20d7=r\u20d7\\vec{WX} = \\vec{r}<\/p>\n\n\n\n

            (Alternatively, if XY is the top, WX is the base, and opposite to YZ.)<\/p>\n\n\n\n

            \n\n\n\n

            d) WA\u20d7\\vec{WA}<\/h3>\n\n\n\n

            Since A trisects XY, and lies one-third of the way from X to Y: XA\u20d7=13XY\u20d7=13r\u20d7\\vec{XA} = \\frac{1}{3} \\vec{XY} = \\frac{1}{3} \\vec{r}<\/p>\n\n\n\n

            From W to A:<\/p>\n\n\n\n

              \n
            • WA\u20d7=WX\u20d7+XA\u20d7=r\u20d7+13r\u20d7=43r\u20d7\\vec{WA} = \\vec{WX} + \\vec{XA} = \\vec{r} + \\frac{1}{3} \\vec{r} = \\frac{4}{3} \\vec{r}<\/li>\n<\/ul>\n\n\n\n

              Answer<\/strong>:
              WA\u20d7=43r\u20d7\\vec{WA} = \\frac{4}{3} \\vec{r}<\/p>\n\n\n\n

              \n\n\n\n

              Conclusion (Summary):<\/h3>\n\n\n\n
                \n
              • a) XY\u20d7=r\u20d7\\vec{XY} = \\vec{r}<\/li>\n\n\n\n
              • b) CY\u20d7=r\u20d7\u221232m\u20d7\\vec{CY} = \\vec{r} – \\frac{3}{2} \\vec{m}<\/li>\n\n\n\n
              • c) WX\u20d7=r\u20d7\\vec{WX} = \\vec{r}<\/li>\n\n\n\n
              • d) WA\u20d7=43r\u20d7\\vec{WA} = \\frac{4}{3} \\vec{r}<\/li>\n<\/ul>\n\n\n\n

                These expressions help in vector geometry by breaking complex paths into combinations of known vectors.<\/p>\n\n\n\n

                \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

                WXYZ is a parallelogram. In the diagram, C is the midpoint of WZ, and A and B trisect XY. Express each vector in terms of a linear combination of r m and . a) XY b)CY c)WX d)WA The Correct Answer and Explanation is: The image you provided shows a parallelogram ABCD, with diagonals intersecting […]<\/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-221533","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221533","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=221533"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221533\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221533"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221533"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}