Miscellaneous Exercise, Question 1
Find the angle between lines with direction ratios \(a, b, c\) and \(b - c, c - a, a - b\).
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- \(a(b - c) + b(c - a) + c(a - b) = 0\).
Answer: \(90^\circ\)
NCERT Solutions for Class 12 Maths Chapter 11 Miscellaneous Exercise
The Miscellaneous Exercise on Three Dimensional Geometry. Perpendicular lines: \(a_1a_2 + b_1b_2 + c_1c_2 = 0\). A line perpendicular to two given lines is parallel to \(\vec b_1 \times \vec b_2\).
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Miscellaneous Exercise questions and solutions
Miscellaneous Exercise, Question 2
Find the equation of the line through the origin parallel to the x-axis.
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- Point \((0, 0, 0)\), direction \(\hat i\) (direction ratios \(1, 0, 0\)).
Answer: \(\vec r = \lambda\hat i\), i.e. \[\begin{aligned}\dfrac{x}{1} &= \dfrac{y}{0} \\ &= \dfrac{z}{0}\end{aligned}\]
Miscellaneous Exercise, Question 3
The lines \(\dfrac{x - 1}{-3} = \dfrac{y - 2}{2k} = \dfrac{z - 3}{2}\) and \(\dfrac{x - 1}{3k} = \dfrac{y - 1}{1} = \dfrac{z - 6}{-5}\) are perpendicular. Find k.
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- \(-3(3k) + 2k(1) + 2(-5) = -7k - 10 = 0\).
Answer: \(k = -\tfrac{10}{7}\)
Miscellaneous Exercise, Question 4
Find the shortest distance between \(\vec r = 6\hat i + 2\hat j + 2\hat k + \lambda(\hat i - 2\hat j + 2\hat k)\) and \(\vec r = -4\hat i - \hat k + \mu(3\hat i - 2\hat j - 2\hat k)\).
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- \[\vec b_1 \times \vec b_2 = 8\hat i + 8\hat j + 4\hat k\], magnitude 12; \[\vec a_2 - \vec a_1 = -10\hat i - 2\hat j - 3\hat k\]
- \(d = \dfrac{|-80 - 16 - 12|}{12}\).
Answer: 9 units
Miscellaneous Exercise, Question 5
Find the vector equation of the line through \((1, 2, -4)\) perpendicular to \(\dfrac{x - 8}{3} = \dfrac{y + 19}{-16} = \dfrac{z - 10}{7}\) and \(\dfrac{x - 15}{3} = \dfrac{y - 29}{8} = \dfrac{z - 5}{-5}\).
Show solution
- Direction: \[\begin{aligned}(3\hat i - 16\hat j + 7\hat k) \times (3\hat i + 8\hat j - 5\hat k) &= 24\hat i + 36\hat j + 72\hat k \\ &= 12(2\hat i + 3\hat j + 6\hat k)\end{aligned}\]
Answer: \[\vec r = (\hat i + 2\hat j - 4\hat k) + \lambda(2\hat i + 3\hat j + 6\hat k)\]
Done the NCERT exercises? The board paper asks more
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.
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).
Also useful: free MCQs and case studies for Three Dimensional Geometry · formulas for this chapter (Class 12 formula sheet, free PDF) · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
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