{"id":6114,"date":"2021-10-07T17:54:22","date_gmt":"2021-10-07T12:24:22","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6114"},"modified":"2021-10-09T00:44:20","modified_gmt":"2021-10-08T19:14:20","slug":"angle-between-two-lines-in-3d","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/angle-between-two-lines-in-3d\/","title":{"rendered":"Angle Between Two Lines in 3d"},"content":{"rendered":"

Here you learn formula for angle between two lines in 3d in both vector form and cartesian form with examples.<\/p>\n

Let’s begin – <\/p>\n

Angle Between Two Lines in 3d<\/h2>\n

(a) Vector Form <\/h3>\n

Let the vector equations of the two lines be \\(\\vec{r}\\) = \\(\\vec{a_1}\\) + \\(\\lambda \\vec{b_1}\\) and \\(\\vec{r}\\) = \\(\\vec{a_2}\\) + \\(\\mu \\vec{b_2}\\).<\/p>\n

These two lines are parallel to the vectors \\(\\vec{b_1}\\) and \\(\\vec{b_2}\\) respectively. Therefore, angle between  these two lines is equal to the angle between \\(\\vec{b_1}\\) and \\(\\vec{b_2}\\).<\/p>\n

Thus, if \\(\\theta\\) is the angle between the given lines, then<\/p>\n

\n

\\(cos\\theta\\) = \\(\\vec{b_1}.\\vec{b_2}\\over |\\vec{b_1}||\\vec{b_2}|\\)<\/p>\n<\/blockquote>\n

Condition of Perpendicularity<\/strong> : If the lines \\(\\vec{b_1}\\) and \\(\\vec{b_2}\\) are perpendicular. Then<\/p>\n

\n

\\(\\vec{b_1}\\). \\(\\vec{b_2}\\) = 0<\/p>\n<\/blockquote>\n

Condition of Parallelism<\/strong> : If the lines are parallel, then \\(\\vec{b_1}\\) and \\(\\vec{b_2}\\) are parallel,<\/p>\n

\n

\\(\\therefore\\) \\(\\vec{b_1}\\) = \\(\\lambda \\vec{b_2}\\) for some scalar \\(\\lambda\\)<\/p>\n<\/blockquote>\n

(b) Cartesian Form<\/h3>\n

Let the cartesian equation of the two lines be<\/p>\n

\\(x – x_1\\over a_1\\) = \\(y – y_1\\over b_1\\) = \\(z – z_1\\over c_1\\)                       …………(i)<\/p>\n

and \\(x – x_1\\over a_2\\) = \\(y – y_1\\over b_2\\) = \\(z – z_1\\over c_2\\)                 …………(ii)<\/p>\n

Direction ratios of line (i) are proportional to \\(a_1\\), \\(b_1\\), \\(c_1\\).<\/p>\n

\\(\\therefore\\) \\(\\vec{m_1}\\) = Vector parallel to line (i) = \\(a_1\\hat{i} + b_1\\hat{j} + c_1\\hat{k}\\).<\/p>\n

Direction ratios of line (ii) are proportional to \\(a_2\\), \\(b_2\\), \\(c_2\\).<\/p>\n

\\(\\therefore\\) \\(\\vec{m_2}\\) = Vector parallel to line (ii) = \\(a_2\\hat{i} + b_2\\hat{j} + c_2\\hat{k}\\).<\/p>\n

Let \\(\\theta\\) be the angle between (i) and (ii).<\/p>\n

Then, \\(\\theta\\) is also the angle between \\(\\vec{m_1}\\) and \\(\\vec{m_2}\\).<\/p>\n

\n

\\(\\therefore\\) \\(cos\\theta\\) = \\(\\vec{m_1}.\\vec{m_2}\\over |\\vec{m_1}||\\vec{m_2}|\\)<\/p>\n

\\(\\implies\\) \\(cos\\theta\\) = \\(a_1a_2 + b_1b_2 + c_1c_2\\over \\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\\)<\/p>\n<\/blockquote>\n

Condition of Perpendicularity<\/strong> : If the lines are perpendicular. Then<\/p>\n

\n

\\(\\vec{m_1}\\). \\(\\vec{m_2}\\) = 0 \\(\\implies\\) \\(a_1a_2 + b_1b_2 + c_1c_2\\) = 0<\/p>\n<\/blockquote>\n

Condition of Parallelism<\/strong> : If the lines are parallel, then \\(\\vec{m_1}\\) and \\(\\vec{m_2}\\) are parallel,<\/p>\n

\n

\\(\\therefore\\) \\(\\vec{m_1}\\) = \\(\\lambda \\vec{m_2}\\) for some scalar \\(\\lambda\\)<\/p>\n

\\(\\implies\\) \\(a_1\\over a_2\\) = \\(b_1\\over b_2\\) = \\(c_1\\over c_2\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the angle between the lines <\/p>\n

\\(\\vec{r}\\) = \\(3\\hat{i} + 2\\hat{j} – 4\\hat{k}\\) + \\(\\lambda\\) (\\(\\hat{i} + 2\\hat{j} + 2\\hat{k}\\)) and<\/p>\n

\\(\\vec{r}\\) = (\\(5\\hat{j} – 2\\hat{k}\\)) + \\(\\mu\\) (\\(3\\hat{i} + 2\\hat{j} + 6\\hat{k}\\))<\/p>\n

Solution<\/span><\/strong> : Let \\(\\theta\\) be the angle between the given lines. These given lines are parallel to the vectors \\(\\vec{b_1}\\) = \\(\\hat{i} + 2\\hat{j} + 2\\hat{k}\\) and \\(\\vec{b_2}\\) = \\(3\\hat{i} + 2\\hat{j} + 6\\hat{k}\\) respectively.<\/p>\n

So, the angle \\(\\theta\\) between them is given by<\/p>\n

\\(cos\\theta\\) = \\(\\vec{b_1}.\\vec{b_2}\\over |\\vec{b_1}||\\vec{b_2}|\\)<\/p>\n

\\(\\implies\\) \\(cos\\theta\\) = \\(3 + 4 + 12\\over \\sqrt{1 + 4 + 4}\\sqrt{9 + 4 + 36}\\) = \\(19\\over 21\\)<\/p>\n

Hence, \\(\\theta\\) = \\(cos^{-1}({19\\over 21})\\)<\/p>\n\n\n

\n
Next – Point of Intersection of Two lines in 3d<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Reduction of Cartesian Form of Line to Vector Form and Vice-Versa<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you learn formula for angle between two lines in 3d in both vector form and cartesian form with examples. Let’s begin –  Angle Between Two Lines in 3d (a) Vector Form  Let the vector equations of the two lines be \\(\\vec{r}\\) = \\(\\vec{a_1}\\) + \\(\\lambda \\vec{b_1}\\) and \\(\\vec{r}\\) = \\(\\vec{a_2}\\) + \\(\\mu \\vec{b_2}\\). These …<\/p>\n

Angle Between Two Lines in 3d<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[33],"tags":[],"yoast_head":"\nAngle Between Two Lines in 3d - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you learn formula for angle between two lines in 3d in both vector form and cartesian form with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/angle-between-two-lines-in-3d\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Angle Between Two Lines in 3d - 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