NCERT Solutions · Class 12 · Chapter 11: Three Dimensional Geometry

NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1

Exercise 11.1: Direction cosines and direction ratios. 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 \(x_2 - x_1, y_2 - y_1, z_2 - z_1\).

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Exercise 11.1 questions and solutions

Exercise 11.1, Question 1

A line makes angles \(90^\circ, 135^\circ, 45^\circ\) with the x, y, z axes. Find its direction cosines.

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\(\cos 90^\circ = 0\), \(\cos 135^\circ = -\tfrac{1}{\sqrt2}\), \(\cos 45^\circ = \tfrac{1}{\sqrt2}\).

Answer: \(0, -\tfrac{1}{\sqrt2}, \tfrac{1}{\sqrt2}\)

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Exercise 11.1, Question 2

Find the direction cosines of a line making equal angles with the coordinate axes.

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\(l = m = n\) and \(3l^2 = 1\).

Answer: \[\pm\tfrac{1}{\sqrt3}, \pm\tfrac{1}{\sqrt3}, \pm\tfrac{1}{\sqrt3}\] (all the same sign)

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Exercise 11.1, Question 3

A line has direction ratios \(-18, 12, -4\). Find its direction cosines.

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\(\sqrt{324 + 144 + 16} = 22\).

Answer: \[-\tfrac{9}{11}, \tfrac{6}{11}, -\tfrac{2}{11}\] (or all signs reversed)

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Exercise 11.1, Question 4

Show \((2, 3, 4)\), \((-1, -2, 1)\), \((5, 8, 7)\) are collinear.

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AB has direction ratios \(-3, -5, -3\); BC has \(6, 10, 6 = -2(-3, -5, -3)\): parallel, with B common.

Answer: Shown

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Exercise 11.1, Question 5

Find the direction cosines of the sides of the triangle with vertices \((3, 5, -4)\), \((-1, 1, 2)\), \((-5, -5, -2)\).

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AB: \(-4, -4, 6\), length \(2\sqrt{17}\). BC: \(-4, -6, -4\), length \(2\sqrt{17}\). CA: \(8, 10, -2\), length \(2\sqrt{42}\).

Answer: AB: \[-\tfrac{2}{\sqrt{17}}, -\tfrac{2}{\sqrt{17}}, \tfrac{3}{\sqrt{17}}\]; BC: \[-\tfrac{2}{\sqrt{17}}, -\tfrac{3}{\sqrt{17}}, -\tfrac{2}{\sqrt{17}}\]; CA: \[\tfrac{4}{\sqrt{42}}, \tfrac{5}{\sqrt{42}}, -\tfrac{1}{\sqrt{42}}\]

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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.

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Three Dimensional Geometry in our sample papers: Sample paper 1 (questions 14, 19, 30, 35) · Sample paper 2 (questions 17, 18, 25, 30) · Sample paper 3 (questions 15, 19, 30, 35) · Sample paper 4 (questions 17, 18, 30, 34) · Sample paper 5 (questions 14, 15, 30, 35).

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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.