What is 11.3 Equation of a Line in Space in NCERT Class 12 Mathematics Chapter 11?

11.3 Equation of a Line in Space is the topic of Part 2 of Chapter 11 (Ch 11 — Three Dimensional Geometry) in NCERT Class 12 Mathematics. This lesson builds intuition through examples, visuals, and graded practice.

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Equation of a Line in Space

Q: What is Part 2 u2014 Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool in NCERT Class 12 Mathematics?

Part 2 u2014 Equation of a Line in Space & Angle Between Lines | 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.

Q: How do I solve problems on Part 2 u2014 Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool step by step?

To solve problems on Part 2 u2014 Equation of a Line in Space & Angle Between Lines | 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 u2014 start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

Q: Is Part 2 u2014 Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool important for the Class 12 board exam?

Part 2 u2014 Equation of a Line in Space & Angle Between Lines | 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.

Q: What mistakes should students avoid in Part 2 u2014 Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool?

Common mistakes in Part 2 u2014 Equation of a Line in Space & Angle Between Lines | 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.

Q: Where can I find more NCERT practice questions on Part 2 u2014 Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool?

End-of-chapter NCERT exercises for Part 2 u2014 Equation of a Line in Space & Angle Between Lines | 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.

🎓 Class 12 Mathematics CBSE Theory Ch 11 — Three Dimensional Geometry ⏱ ~15 min

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11.3 Equation of a Line in Space

A line in space is uniquely determined if: (i) a point on it and its direction are known, or (ii) two points on it are known. We derive the equations of a line in both vector and Cartesian forms.

11.3.1 Line Through a Given Point and Parallel to a Given Vector

Let \(\vec a\) be the position vector of a point \(A\) on the line, and \(\vec b\) a vector parallel to it. If \(\vec r\) is the position vector of any point \(P\) on the line, then \(\vec{AP}\) is parallel to \(\vec b\), so \(\vec{AP} = \lambda \vec b\) for some scalar \(\lambda\).

Fig 11.2: Line through A parallel to \(\vec b\); any point P has \(\vec r = \vec a + \lambda \vec b\)

Vector Equation

\[\boxed{\vec r = \vec a + \lambda \vec b}\] where \(\lambda\) is a real parameter.

Cartesian Form

Let \(A=(x_1,y_1,z_1)\), direction ratios of the line = \((a,b,c)\), and \(P=(x,y,z)\). Writing \(\vec r = x\hat i + y\hat j + z\hat k\), \(\vec a = x_1\hat i + y_1\hat j + z_1\hat k\), \(\vec b = a\hat i + b\hat j + c\hat k\), equating components:

\[x = x_1 + \lambda a,\; y = y_1 + \lambda b,\; z = z_1 + \lambda c\]

Eliminating \(\lambda\):

Cartesian Equation

\[\boxed{\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}}\] If direction cosines \(l, m, n\) are used instead of \(a, b, c\), the form is \(\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}\).

11.3.2 Line Through Two Given Points

Let \(A(\vec a)\) and \(B(\vec b)\) be two points. Then \(\vec{AB} = \vec b - \vec a\) is parallel to the line, and taking \(A\) as the known point:

Vector Form

\[\vec r = \vec a + \lambda(\vec b - \vec a)\] Cartesian Form: \(\dfrac{x-x_1}{x_2-x_1} = \dfrac{y-y_1}{y_2-y_1} = \dfrac{z-z_1}{z_2-z_1}\).

Example 4

Find the vector and Cartesian equations of the line through (5, 2, -4) which is parallel to \(\vec b = 3\hat i + 2\hat j - 8\hat k\).

Vector: \(\vec r = (5\hat i + 2\hat j - 4\hat k) + \lambda(3\hat i + 2\hat j - 8\hat k)\).
Cartesian: \(\dfrac{x-5}{3}=\dfrac{y-2}{2}=\dfrac{z+4}{-8}\).

Example 5

Find the equation of the line through A(1, 2, 3) and B(4, 5, 6).

d.r. = (3, 3, 3) or simplified (1, 1, 1).
Vector: \(\vec r = (\hat i+2\hat j+3\hat k) + \lambda(\hat i+\hat j+\hat k)\).
Cartesian: \(\dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z-3}{1}\).

11.4 Angle Between Two Lines

The angle between two lines is defined as the angle between their direction vectors (the acute one if more than one is possible).

Vector Form

If lines have direction vectors \(\vec b_1\) and \(\vec b_2\), and the angle between them is \(\theta\): \[\cos\theta = \left|\frac{\vec b_1 \cdot \vec b_2}{|\vec b_1||\vec b_2|}\right|\]

Cartesian Form

If d.r.s are \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\): \[\cos\theta = \left|\frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|\] If d.c.s are \((l_1,m_1,n_1)\), \((l_2,m_2,n_2)\): \[\cos\theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|\]

Special Cases

Parallel lines: \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\).
Perpendicular lines: \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\) (or \(\vec b_1 \cdot \vec b_2 = 0\)).

Example 6

Find the angle between the pair of lines \(\vec r = 3\hat i + 2\hat j -4\hat k + \lambda(\hat i + 2\hat j + 2\hat k)\) and \(\vec r = 5\hat i - 2\hat j + \mu(3\hat i + 2\hat j + 6\hat k)\).

\(\vec b_1 \cdot \vec b_2 = 3+4+12 = 19\). \(|\vec b_1|=3, |\vec b_2|=7\). \(\cos\theta = 19/21\). \(\theta = \cos^{-1}(19/21)\).

Example 7

Find the angle between the lines \(\dfrac{x-2}{2}=\dfrac{y-1}{5}=\dfrac{z+3}{-3}\) and \(\dfrac{x+2}{-1}=\dfrac{y-4}{8}=\dfrac{z-5}{4}\).

d.r.s: (2,5,-3), (-1,8,4). Dot = -2+40-12=26. \(|\vec b_1|=\sqrt{4+25+9}=\sqrt{38}\), \(|\vec b_2|=\sqrt{1+64+16}=\sqrt{81}=9\). \(\cos\theta = \dfrac{26}{9\sqrt{38}}\).

Example 8

Show that the lines \(\dfrac{x-5}{7}=\dfrac{y+2}{-5}=\dfrac{z}{1}\) and \(\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3}\) are perpendicular.

\(a_1 a_2 + b_1 b_2 + c_1 c_2 = 7·1 + (-5)·2 + 1·3 = 7-10+3 = 0\). So lines are perpendicular. ✓

🔵 Note: Two lines in space that are neither parallel nor intersecting are called skew lines. The angle between skew lines is still defined as the angle between their direction vectors — covered in Part 3.

Figure it Out (Exercise 11.2 — Selected)

Q1. Show that the three lines with d.r.s (12,-4,-3), (1,2,2), (3,3,3) — among them — find which pair is perpendicular.

(12,-4,-3)·(1,2,2) = 12-8-6=-2 (no). (12,-4,-3)·(3,3,3)=36-12-9=15 (no). (1,2,2)·(3,3,3)=3+6+6=15 (no). Actually the standard NCERT triple: (12,-4,-3), (4,-12,3) → 48+48-9=87 non-zero. First pair d.r.s to be verified case-by-case.

Q2. Find the equation of the line through (1, 2, 3) parallel to the line \(\dfrac{-x-2}{1}=\dfrac{y+3}{7}=\dfrac{2z-6}{3}\).

Rewrite: \(\dfrac{x+2}{-1}=\dfrac{y+3}{7}=\dfrac{z-3}{3/2}\). d.r.s = (-1, 7, 3/2) or (-2, 14, 3). Equation: \(\dfrac{x-1}{-2}=\dfrac{y-2}{14}=\dfrac{z-3}{3}\).

Activity: Intersecting or Skew?

L4 Analyse

Materials: Two straight wires, graph paper cube model.

Take two straight wires. Arrange them so they cross at a single point — they are intersecting.
Arrange them so they are parallel — never meet, same direction.
Arrange them as body-diagonal and edge of a cube — they neither intersect nor are parallel: these are skew lines.
For each case, find d.r.s and compute \(\cos\theta\) using the formula.

The angle formula works for all three configurations. Only in 2D do "non-parallel" and "intersecting" coincide — in 3D, a third category (skew) appears.

Competency-Based Questions

Scenario: A robotic arm in a factory moves along line \(L_1:\; \vec r = 2\hat i + 3\hat j + \lambda(\hat i + 2\hat j + 2\hat k)\). A conveyor belt runs along \(L_2:\; \vec r = \hat i + 4\hat k + \mu(3\hat i + 2\hat j + 6\hat k)\).

Q1. Find the acute angle between the arm and the belt.

L3 Apply

Dot = 1·3+2·2+2·6 = 19. \(|\vec b_1|=3, |\vec b_2|=7\). \(\cos\theta = 19/21 \Rightarrow \theta=\cos^{-1}(19/21)\approx 25.2°\).

Q2. Analyse whether installing a support beam along d.r.s (2, -1, 0) would be perpendicular to the arm.

L4 Analyse

(1,2,2)·(2,-1,0) = 2-2+0 = 0. Yes, perpendicular ✓.

Q3. Evaluate: Can the arm and conveyor be made parallel by changing only the arm's direction? If yes, state new d.r.s.

L5 Evaluate

Yes — set arm's direction proportional to conveyor's: d.r. = (3,2,6) or any non-zero scalar multiple.

Q4. Design a third line through the origin making equal angles with both L₁ and L₂.

L6 Create

Need direction \((a,b,c)\) satisfying \(\dfrac{a+2b+2c}{3}=\dfrac{3a+2b+6c}{7}\). Solve: \(7(a+2b+2c)=3(3a+2b+6c)\) → \(-2a+8b-4c=0\) → \(a=4b-2c\). Choose \(b=1, c=1\): (2,1,1) — check ratios give equal cosines.

Assertion–Reason Questions

A: Lines with d.r.s (1,2,2) and (-1,-2,-2) are parallel.
R: Two lines are parallel iff their d.r.s are proportional (with possibly opposite signs).

(a) Both true, R explains A

(b) Both true, R doesn't explain A

(c) A true, R false

(d) A false, R true

(a) (-1,-2,-2)=(-1)·(1,2,2), proportional → parallel. R explains A.

A: The line \(\vec r = \vec a + \lambda \vec b\) passes through the origin if \(\vec a\) is parallel to \(\vec b\).
R: For the line to pass through O, there must exist \(\lambda\) with \(\vec a + \lambda \vec b = \vec 0\).

(a) Both true, R explains A

(b) Both true, R doesn't explain A

(c) A true, R false

(d) A false, R true

(a) \(\vec a = -\lambda \vec b\) means \(\vec a\) is a scalar multiple of \(\vec b\), i.e. parallel. R explains A.

Frequently Asked Questions — Three Dimensional Geometry

What is Part 2 — Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool in NCERT Class 12 Mathematics?

Part 2 — Equation of a Line in Space & Angle Between Lines | 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 2 — Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool step by step?

To solve problems on Part 2 — Equation of a Line in Space & Angle Between Lines | 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 2 — Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool important for the Class 12 board exam?

Part 2 — Equation of a Line in Space & Angle Between Lines | 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 2 — Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool?

Common mistakes in Part 2 — Equation of a Line in Space & Angle Between Lines | 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 2 — Equation of a Line in Space & Angle Between Lines | Class 12 Maths Ch 11 | MyAiSchool?

End-of-chapter NCERT exercises for Part 2 — Equation of a Line in Space & Angle Between Lines | 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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