Three Dimensional Geometry studies points, lines and planes in space using coordinates and vectors. Important Questions Class 12 Maths Chapter 11 help students practise direction cosines, direction ratios, line equations, angles, shortest distance and PYQ-style 3D geometry questions.
Class 12 Maths Chapter 11 becomes easier when students connect every question with one idea: the direction of a line in space. These Important Questions Class 12 Maths Chapter 11 are arranged from basic direction cosines to equation of line, angle between lines, shortest distance and 5-mark practice. Students should revise vectors class 12 concepts first because dot product, cross product and vector equations are used throughout Three Dimensional Geometry.
Key Takeaways from Important Questions Class 12 Maths Chapter 11
| Detail |
Information |
| Chapter Name |
Three Dimensional Geometry |
| Chapter Number |
Chapter 11 |
| Class |
Class 12 |
| Subject |
Mathematics |
| Main Tools |
Direction cosines, direction ratios and vectors |
| Important Question Types |
Lines, angles, shortest distance and Cartesian-vector conversion |
| High-Scoring Area |
Shortest distance between skew lines |
| Common Mistake |
Mixing direction ratios with direction cosines |
| Best Revision Method |
Formula table, solved examples and PYQ-style practice |
3D Geometry Class 12 Important Questions: Chapter Overview
3D geometry class 12 important questions mainly come from lines in space.
Students should know how to find the direction of a line, write its equation, check relation between two lines and calculate shortest distance.
The chapter covers these core areas:
- Direction cosines of a line
- Direction ratios of a line
- Direction ratios from two points
- Equation of line in vector form
- Equation of line in Cartesian form
- Angle between two lines
- Parallel and perpendicular lines
- Skew lines
- Shortest distance between skew lines
- Distance between parallel lines
Important Questions Class 12 Maths Chapter-Wise
Three Dimensional Geometry Class 12 Important Questions: Formula Sheet
Three dimensional geometry class 12 important questions become easier when formulas are revised first.
Most mistakes happen because students choose the wrong formula or miss the modulus sign.
| Concept |
Formula |
| Direction cosine condition |
l² + m² + n² = 1 |
| Direction cosines from direction ratios |
l = a / √(a² + b² + c²), m = b / √(a² + b² + c²), n = c / √(a² + b² + c²) |
| Direction ratios of line through two points |
x₂ - x₁, y₂ - y₁, z₂ - z₁ |
| Vector form of line |
r = a + λb |
| Cartesian form of line |
(x - x₁)/a = (y - y₁)/b = (z - z₁)/c |
| Angle between two lines |
cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / √(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²) |
| Perpendicular lines |
a₁a₂ + b₁b₂ + c₁c₂ = 0 |
| Parallel lines |
a₁/a₂ = b₁/b₂ = c₁/c₂ |
3D Important Questions Class 12: Very Short Answer Questions
3D important questions class 12 often start with direction cosines, direction ratios and basic line concepts.
These questions are useful for 1-mark and quick revision.
| Question |
Answer |
| What are direction cosines? |
Direction cosines are the cosines of angles made by a directed line with the positive coordinate axes. |
| What are direction ratios? |
Direction ratios are numbers proportional to the direction cosines of a line. |
| What is the condition for direction cosines l, m, n? |
l² + m² + n² = 1 |
| What are direction ratios of x-axis? |
1, 0, 0 |
| What are direction cosines of y-axis? |
0, 1, 0 |
| What are skew lines? |
Skew lines are non-parallel, non-intersecting lines in space. |
| What is the shortest distance between intersecting lines? |
The shortest distance is zero. |
| What is the vector equation of a line? |
r = a + λb |
Direction Cosines and Direction Ratios Class 12 Questions
Direction cosines and direction ratios class 12 questions form the base of Chapter 11.
These questions usually appear as direct concept questions or as first steps in longer line-equation sums.
Important Questions of 3D Geometry Class 12 on Direction Values
Q1. If a line makes angles 90°, 60° and 30° with the positive directions of x, y and z axes, find its direction cosines.
| Axis |
Angle |
Direction Cosine |
| x-axis |
90° |
0 |
| y-axis |
60° |
1/2 |
| z-axis |
30° |
√3/2 |
Answer: The direction cosines are 0, 1/2, √3/2.
Q2. Find the direction cosines of a line having direction ratios 2, -1, -2.
Magnitude factor = √(2² + (-1)² + (-2)²)
= √9
= 3
| Direction Ratio |
Direction Cosine |
| 2 |
2/3 |
| -1 |
-1/3 |
| -2 |
-2/3 |
Answer: The direction cosines are 2/3, -1/3, -2/3.
Q3. Find the direction ratios of the line joining A(2, 3, 4) and B(5, 7, 9).
Direction ratios = x₂ - x₁, y₂ - y₁, z₂ - z₁
| Difference |
Value |
| 5 - 2 |
3 |
| 7 - 3 |
4 |
| 9 - 4 |
5 |
Answer: The direction ratios are 3, 4, 5.
Equation of Line in 3D Class 12 Questions
Equation of line in 3D class 12 questions usually ask for vector form, Cartesian form or conversion between the two.
A line in space becomes easy when students identify one point and one direction vector.
Class 12 Maths 3D Geometry Important Questions on Line Equations
Q4. Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to 3i + 2j - 2k.
The given point is (1, 2, 3).
The direction vector is 3i + 2j - 2k.
| Form |
Equation |
| Vector form |
r = i + 2j + 3k + λ(3i + 2j - 2k) |
| Cartesian form |
(x - 1)/3 = (y - 2)/2 = (z - 3)/-2 |
Answer: The vector and Cartesian equations are given above.
Q5. Find the Cartesian equation of the line passing through (-2, 4, -5) and parallel to the line (x + 3)/3 = (y - 4)/5 = (z + 8)/6.
The required line has the same direction ratios as the given line.
| Given Point |
Direction Ratios |
| (-2, 4, -5) |
3, 5, 6 |
Answer: The Cartesian equation is:
(x + 2)/3 = (y - 4)/5 = (z + 5)/6
3D Geometry Class 12 PYQ-Style Questions on Angle Between Lines
3D geometry class 12 PYQ-style questions often test the angle between two lines.
These sums use direction ratios and the dot product formula.
Three Dimensional Geometry Class 12 PYQ Practice on Angles
Q6. Find the angle between the lines with direction ratios 3, 5, 4 and 1, 1, 2.
| Step |
Value |
| a₁a₂ + b₁b₂ + c₁c₂ |
3(1) + 5(1) + 4(2) = 16 |
| √(a₁² + b₁² + c₁²) |
√(9 + 25 + 16) = √50 |
| √(a₂² + b₂² + c₂²) |
√(1 + 1 + 4) = √6 |
cos θ = 16 / √300
cos θ = 8 / 5√3
Answer: θ = cos⁻¹(8 / 5√3)
Q7. Show that the lines with direction ratios 2, -1, 1 and 1, 3, 1 are perpendicular.
For perpendicular lines:
a₁a₂ + b₁b₂ + c₁c₂ = 0
| Calculation |
Value |
| 2(1) + (-1)(3) + 1(1) |
2 - 3 + 1 = 0 |
Answer: The lines are perpendicular.
Shortest Distance Between Skew Lines Class 12 Questions
Shortest distance between skew lines class 12 is one of the most important 3D Geometry topics.
Skew lines never meet and are not parallel. Their shortest distance lies along a direction perpendicular to both lines.
Class 12 Maths 3D Important Questions on Skew Lines
Q8. Find the shortest distance between the lines r = i + 2j + k + λ(i - j + k) and r = 2i - j - k + μ(2i + j + 2k).
| Element |
Value |
| a₁ |
i + 2j + k |
| b₁ |
i - j + k |
| a₂ |
2i - j - k |
| b₂ |
2i + j + 2k |
| a₂ - a₁ |
i - 3j - 2k |
Formula:
Shortest distance = |(b₁ × b₂) · (a₂ - a₁)| / |b₁ × b₂|
Now,
b₁ × b₂ = -3i + 3k
(b₁ × b₂) · (a₂ - a₁)
= (-3i + 3k) · (i - 3j - 2k)
= -3 - 6
= -9
| (b₁ × b₂) · (a₂ - a₁) | = 9
| b₁ × b₂ | = √(9 + 9)
= 3√2
Shortest distance = 9 / 3√2
= 3 / √2
Answer: The shortest distance is 3/√2 units.
Q9. What is the shortest distance between two intersecting lines?
Answer: The shortest distance between two intersecting lines is zero because the lines meet at one common point.
Q10. What is the shortest distance between two parallel lines?
Answer: The shortest distance between two parallel lines is the perpendicular distance from any point on one line to the other line.
3D Geometry Class 12 5 Mark Questions
3D geometry class 12 5 mark questions usually combine line equations, direction ratios, shortest distance, angle between lines or proof-based reasoning.
Students should show every step because marks are often given for formula, substitution and simplification.
Class 12 3D Geometry Important Questions for 5 Marks
Q11. Find the equation of the line passing through two points A(1, 2, -4) and B(3, 5, -6).
Direction ratios of AB:
| Difference |
Value |
| 3 - 1 |
2 |
| 5 - 2 |
3 |
| -6 - (-4) |
-2 |
Using point A(1, 2, -4), the Cartesian equation is:
(x - 1)/2 = (y - 2)/3 = (z + 4)/-2
The vector equation is:
r = i + 2j - 4k + λ(2i + 3j - 2k)
Answer: The required line equations are given above.
Q12. Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.
| Line Segment |
Direction Ratios |
| AB |
-1, -5, 7 |
| BC |
2, 10, -14 |
The direction ratios of AB and BC are proportional.
AB = (-1, -5, 7)
BC = -2AB
Answer: The points A, B and C are collinear.
Q13. Find the vector equation of the line passing through (5, 2, -4) and parallel to 3i + 2j - 8k.
Vector equation:
r = 5i + 2j - 4k + λ(3i + 2j - 8k)
Cartesian form:
(x - 5)/3 = (y - 2)/2 = (z + 4)/-8
Answer: The vector and Cartesian equations are given above.
3D Geometry Class 12 Extra Questions for Practice
3D geometry class 12 extra questions are useful after students finish NCERT examples and board-style solved questions.
| Question |
Practice Focus |
| Find direction cosines of a line equally inclined to all axes. |
Direction cosines |
| Convert r = 2i - j + 4k + λ(i + 2j - k) into Cartesian form. |
Vector to Cartesian |
| Find the angle between two lines with direction ratios 1, 2, 2 and 3, 2, 6. |
Angle between lines |
| Show that two given lines are parallel if their direction ratios are proportional. |
Parallel lines |
| Find the equation of a line through (1, 2, 3) and parallel to 2i - j + 4k. |
Line equation |
| Find the shortest distance between two skew lines in vector form. |
Skew lines |
| Find the distance between two parallel lines. |
Parallel line distance |
| Prove that three points are collinear using direction ratios. |
Collinearity |
Important Questions from 3D Geometry Class 12: Common Mistakes
Important questions from 3D Geometry Class 12 often use similar-looking formulas.
Students should identify the line type before applying the formula.
| Mistake |
Correct Approach |
| Using direction cosines as direction ratios without checking |
Direction ratios only need proportional values |
| Forgetting modulus in angle formula |
Use absolute value for acute angle |
| Mixing vector and Cartesian forms |
Convert step by step |
| Taking wrong point for line equation |
Use the point given in the question |
| Confusing skew and parallel lines |
Check proportional direction ratios first |
| Skipping cross product in shortest distance |
Use b₁ × b₂ for skew lines |
Three Dimensional Geometry Class 12 PYQ Practice Strategy
Three dimensional geometry class 12 PYQ practice works best after formula revision.
Students should not begin with difficult 5-mark questions. They should move from direction values to line equations and then shortest distance.
| Stage |
What to Practise |
| First |
Direction cosines and direction ratios |
| Next |
Equation of line in vector and Cartesian form |
| Then |
Angle between two lines |
| After that |
Parallel and perpendicular line checks |
| Final |
Shortest distance between skew lines |
PYQs help students understand board question style, marking steps and common formula use.
Quick Revision Notes for Three Dimensional Geometry Class 12
Three Dimensional Geometry uses vectors to study lines in space.
Direction cosines show the angles a line makes with the coordinate axes. Direction ratios are proportional to direction cosines and help write line equations.
The equation of a line can appear in vector form or Cartesian form. Students should know how to convert one form into the other.
The angle between two lines uses direction ratios or direction cosines. The shortest distance formula is mainly used for skew lines.