Direction cosines and direction ratios of a line, equations of a line in vector and cartesian form, the angle between two lines, and the shortest distance between two lines. Our own step-by-step solutions to Exercises 11.1, 11.2 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. A line at angles \(\alpha, \beta, \gamma\) to the axes has direction cosines \(l = \cos\alpha\), \(m = \cos\beta\), \(n = \cos\gamma\), with \(l^2 + m^2 + n^2 = 1\). Direction ratios a, b, c are any numbers proportional to them: \(l = \dfrac{\pm a}{\sqrt{a^2 + b^2 + c^2}}\) and so on. Through \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) the direction ratios are \(x_2 - x_1, y_2 - y_1, z_2 - z_1\).
What it tests. Line through \(\vec a\) parallel to \(\vec b\): \(\vec r = \vec a + \lambda\vec b\), i.e. \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}\). Angle between lines: \(\cos\theta = \dfrac{|\vec b_1 \cdot \vec b_2|}{|\vec b_1||\vec b_2|}\). Shortest distance between skew lines: \(\left|\dfrac{(\vec b_1 \times \vec b_2) \cdot (\vec a_2 - \vec a_1)}{|\vec b_1 \times \vec b_2|}\right|\); for parallel lines use \(\dfrac{|\vec b \times (\vec a_2 - \vec a_1)|}{|\vec b|}\).
What it tests. Perpendicular lines: \(a_1a_2 + b_1b_2 + c_1c_2 = 0\). A line perpendicular to two given lines is parallel to \(\vec b_1 \times \vec b_2\).
Three Dimensional Geometry has 40 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Three Dimensional Geometry · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.