{"id":133951,"date":"2023-03-12T17:07:45","date_gmt":"2023-03-12T11:37:45","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=133951"},"modified":"2023-03-12T17:07:45","modified_gmt":"2023-03-12T11:37:45","slug":"find-the-distance-between-the-lines-r-i2j-4ky2i3j6k-r-3i3j-5ku4i6j12k","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/find-the-distance-between-the-lines-r-i2j-4ky2i3j6k-r-3i3j-5ku4i6j12k","title":{"rendered":"find the distance between the lines: r =(i+2j-4k)+Y(2i+3j+6k): r =(3i+3j-5k)+u(4i+6j+12k)"},"content":{"rendered":"<h2>Question: find the distance between the lines: r =(i+2j-4k)+Y(2i+3j+6k): r =(3i+3j-5k)+u(4i+6j+12k)<\/h2>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h3>The correct answer is &#8211;<\/h3>\n<div class=\"group w-full text-gray-800 dark:text-gray-100 border-b border-black\/10 dark:border-gray-900\/50 bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0 m-auto\">\n<div class=\"relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<h4>To find the distance between the two skew lines, we need to find a vector connecting the two lines, and then find the projection of the vector connecting any point on one line to the other line.<\/h4>\n<h4>Let&#8217;s first find a vector connecting the two lines. We can take any two points, one on each line, and subtract their position vectors. Let&#8217;s take the point (1, 2, -4) on the first line and the point (3, 3, -5) on the second line:<\/h4>\n<h4>v = (3i + 3j &#8211; 5k) &#8211; (i + 2j &#8211; 4k) = 2i + j &#8211; k<\/h4>\n<h4>Next, we need to find the projection of this vector onto a direction vector of one of the lines. Let&#8217;s take the direction vector of the first line, which is (2i + 3j + 6k). We can use the projection formula:<\/h4>\n<h4>proj_w v = ((v . w) \/ ||w||^2) w<\/h4>\n<h4>where w is the direction vector of the first line.<\/h4>\n<h4>First, we need to find the dot product v . w:<\/h4>\n<h4>v . w = (2i + j &#8211; k) . (2i + 3j + 6k) = 4 + 3 &#8211; 6 = 1<\/h4>\n<h4>Next, we need to find the magnitude of w:<\/h4>\n<h4>||w|| = sqrt((2^2) + (3^2) + (6^2)) = sqrt(49) = 7<\/h4>\n<h4>Finally, we can plug in these values into the projection formula:<\/h4>\n<h4>proj_w v = ((v . w) \/ ||w||^2) w = (1 \/ (7^2)) (2i + 3j + 6k) = (2\/49)i + (3\/49)j + (6\/49)k<\/h4>\n<h4>This projection vector represents the distance between the two lines. The magnitude of this vector is the distance between the two lines:<\/h4>\n<h4>distance = ||proj_w v|| = sqrt((2\/49)^2 + (3\/49)^2 + (6\/49)^2) = sqrt(49)\/49 = 1\/7<\/h4>\n<h4>Therefore, the distance between the two lines is 1\/7.<\/h4>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Question: find the distance between the lines: r =(i+2j-4k)+Y(2i+3j+6k): r =(3i+3j-5k)+u(4i+6j+12k) &nbsp; &nbsp; The correct answer is &#8211; To find the distance between the two skew lines, we need to find a vector connecting the two lines, and then find the projection of the vector connecting any point on one line to the other line. [&hellip;]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-133951","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133951","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=133951"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133951\/revisions"}],"predecessor-version":[{"id":133952,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133951\/revisions\/133952"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=133951"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=133951"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=133951"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}