This MCQ module is based on: Direction Cosines and Direction Ratios
TOPIC 18 OF 25
Direction Cosines and Direction Ratios
🎓 Class 12
Mathematics
CBSE
Theory
Ch 11 — Three Dimensional Geometry
⏱ ~15 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Direction Cosines and Direction Ratios
Targeting Class 12 level in Coordinate Geometry, with Advanced difficulty.
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11.1 Introduction to Three-Dimensional Geometry
In Class XI, we studied points in 3D space using coordinate axes. In this chapter, we extend those ideas to study lines in space using direction cosines? and direction ratios?, and write equations of lines in both vector and Cartesian forms. We will also find the angle between two lines and compute the shortest distance between skew lines.
11.2 Direction Cosines and Direction Ratios of a Line
If a directed line \(L\) passes through the origin and makes angles \(\alpha, \beta, \gamma\) with the positive x, y, z-axes respectively, then \(\cos\alpha, \cos\beta, \cos\gamma\) are called the direction cosines of \(L\), usually denoted by \(l, m, n\).
Fig 11.1: Line OP through origin with direction angles α, β, γ
Key Result
If \(l, m, n\) are the direction cosines of a line, then \[l^2 + m^2 + n^2 = 1\]
Proof: Let \(P(x,y,z)\) be a point on the line at distance \(r\) from O. Then \(x = l r, y = m r, z = n r\), and since \(x^2+y^2+z^2 = r^2\), dividing gives \(l^2+m^2+n^2 = 1\).
Direction Ratios
Any three numbers \(a, b, c\) proportional to direction cosines \(l, m, n\) are called direction ratios. That is, \(\dfrac{a}{l} = \dfrac{b}{m} = \dfrac{c}{n} = k\).
Relation d.r. ↔ d.c.
If \(a, b, c\) are direction ratios, then direction cosines are:
\[l = \pm\frac{a}{\sqrt{a^2+b^2+c^2}},\quad m = \pm\frac{b}{\sqrt{a^2+b^2+c^2}},\quad n = \pm\frac{c}{\sqrt{a^2+b^2+c^2}}\]
The sign depends on the chosen direction of the line.
Direction Cosines of the Line Joining Two Points
Let \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) be two points. Then direction ratios of \(PQ\) are \((x_2-x_1), (y_2-y_1), (z_2-z_1)\), and direction cosines are:
\(l = \dfrac{x_2-x_1}{PQ},\; m = \dfrac{y_2-y_1}{PQ},\; n = \dfrac{z_2-z_1}{PQ}\), where \(PQ = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\).
Example 1
If a line makes angles 90°, 135°, 45° with the positive x, y, z-axes, find its direction cosines.
\(l=\cos 90°=0,\; m=\cos 135°=-\tfrac{1}{\sqrt 2},\; n=\cos 45°=\tfrac{1}{\sqrt 2}\).
Check: \(0 + \tfrac{1}{2} + \tfrac{1}{2} = 1\) ✓
Check: \(0 + \tfrac{1}{2} + \tfrac{1}{2} = 1\) ✓
Example 2
Find the direction cosines of the line passing through A(-2, 4, -5) and B(1, 2, 3).
d.r. = \((1-(-2), 2-4, 3-(-5)) = (3, -2, 8)\).
\(|AB| = \sqrt{9+4+64} = \sqrt{77}\).
d.c. = \(\left(\dfrac{3}{\sqrt{77}}, \dfrac{-2}{\sqrt{77}}, \dfrac{8}{\sqrt{77}}\right)\).
\(|AB| = \sqrt{9+4+64} = \sqrt{77}\).
d.c. = \(\left(\dfrac{3}{\sqrt{77}}, \dfrac{-2}{\sqrt{77}}, \dfrac{8}{\sqrt{77}}\right)\).
Example 3
Show that points A(2,3,-4), B(1,-2,3), C(3,8,-11) are collinear.
d.r. of AB = \((-1,-5,7)\). d.r. of BC = \((2,10,-14) = -2(-1,-5,7)\). Since d.r.s are proportional, AB and BC are parallel; sharing point B, A, B, C are collinear.
🔵 Note: A line can be represented by an infinite set of direction ratios, but only two sets of direction cosines (one for each direction of the line).
Activity: Building a 3D Line Model
L3 ApplyMaterials: Three pencils, clay/putty, thread, protractor.
- Fix three pencils at mutually perpendicular directions using clay, representing x, y, z axes. Call the meeting point O.
- Tie a thread from O to another point P in space.
- Use a protractor to estimate the angles α (with x-axis), β (with y-axis), γ (with z-axis).
- Compute \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma\). Is it close to 1?
- Shift P to new positions. Record how α, β, γ change yet the sum of squares of cosines stays 1.
The identity \(l^2+m^2+n^2=1\) is a geometric statement of the Pythagoras theorem in 3D — the projections of a unit vector on the three axes always form a unit right triangle system.
Figure it Out (Exercise 11.1 — Selected)
Q1. If a line makes angles 90°, 60°, 30° with x, y, z axes respectively, find its direction cosines.
\(l=0, m=\tfrac{1}{2}, n=\tfrac{\sqrt 3}{2}\).
Q2. Find the direction cosines of a line which makes equal angles with the coordinate axes.
If \(\alpha=\beta=\gamma\), then \(3\cos^2\alpha = 1 \Rightarrow \cos\alpha=\pm\tfrac{1}{\sqrt 3}\). So d.c. = \(\left(\pm\tfrac{1}{\sqrt 3},\pm\tfrac{1}{\sqrt 3},\pm\tfrac{1}{\sqrt 3}\right)\).
Q3. If d.r.s of a line are \(-18, 12, -4\), find d.c.s.
\(\sqrt{324+144+16}=\sqrt{484}=22\). d.c. = \(\left(-\tfrac{9}{11}, \tfrac{6}{11}, -\tfrac{2}{11}\right)\).
Competency-Based Questions
Scenario: A drone flies in a straight line from base-station A(0, 0, 0) to delivery-point B(6, 2, 3) (coordinates in km, z = altitude).
Q1. Find the direction cosines of the drone's flight path AB.
L3 Apply\(|AB|=\sqrt{36+4+9}=7\). d.c. = \((6/7, 2/7, 3/7)\).
Q2. Analyse: If the drone must maintain a constant climb angle of \(\cos^{-1}(3/7)\) with the vertical z-axis, verify this is consistent with the path.
L4 AnalyseThe angle with z-axis is \(\gamma\) where \(\cos\gamma = n = 3/7\). So \(\gamma = \cos^{-1}(3/7)\) — consistent ✓.
Q3. Evaluate whether d.r.s (12, 4, 6) represent the same line as AB and justify.
L5 Evaluate(12, 4, 6) = 2·(6, 2, 3). Proportional to AB's d.r.s, so same direction — yes, same line's direction.
Q4. Design a second drone path that is perpendicular to AB and also passes through origin. Give its d.r.s.
L6 CreateNeed \(6a + 2b + 3c = 0\). One choice: (1, 0, -2) — check: 6·1 + 2·0 + 3·(-2) = 0 ✓. Many other valid answers.
Assertion–Reason Questions
A: If d.r.s of a line are (2, -1, 2), its d.c.s are (2/3, -1/3, 2/3).
R: d.c.s are obtained by dividing d.r.s by \(\sqrt{a^2+b^2+c^2}\).
R: d.c.s are obtained by dividing d.r.s by \(\sqrt{a^2+b^2+c^2}\).
(a) \(\sqrt{4+1+4}=3\), dividing gives (2/3,-1/3,2/3).
A: A line making equal angles with all three axes has d.c.s \((1/\sqrt 3,1/\sqrt 3,1/\sqrt 3)\) only.
R: Angles with axes are unique for a directed line.
R: Angles with axes are unique for a directed line.
(d) A false — each coordinate can be + or -, so 8 sign combinations reduce to 2 direction-choices. R true.
A: Points (1,2,3), (2,3,4), (4,5,6) are collinear.
R: If d.r.s of two segments are proportional, the points are collinear.
R: If d.r.s of two segments are proportional, the points are collinear.
(a) d.r.(AB)=(1,1,1), d.r.(BC)=(2,2,2)=2·(1,1,1). Proportional, share B, collinear.
Frequently Asked Questions — Three Dimensional Geometry
What is Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool in NCERT Class 12 Mathematics?
Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool is a key concept covered in NCERT Class 12 Mathematics, Chapter 11: Three Dimensional Geometry. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool step by step?
To solve problems on Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 12 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 11: Three Dimensional Geometry?
The essential formulas of Chapter 11 (Three Dimensional Geometry) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool important for the Class 12 board exam?
Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool is part of the NCERT Class 12 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool?
Common mistakes in Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool?
End-of-chapter NCERT exercises for Part 1 — Direction Cosines & Direction Ratios | Class 12 Maths Ch 11 | MyAiSchool cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 11, and solve at least one previous-year board paper to consolidate your understanding.
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