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Three Dimensional Geometry

NCERT Class 12 · Mathematics Based on NCERT Class 12 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter extends the study of geometry from two dimensions to three dimensions using vector algebra. It covers direction cosines and direction ratios of lines, equations of lines and planes in space, angles between lines and planes, and the shortest distance between skew and parallel lines. The one big idea a student must walk away with is that vector methods simplify 3D geometry, making it elegant and systematic, and that every line and plane can be represented both in vector and Cartesian forms.
Key Concepts (13)
Direction Cosines
If a directed line L passing through the origin makes angles α, β and γ with x, y and z-axes respectively, then cos α, cos β and cos γ are called the direction cosines of the directed line L. They are denoted by l, m, n.
e.g. For a line making angles 90°, 60°, and 30° with the x, y, and z-axes respectively, the direction cosines are 0, 1/2, √3/2.
Direction Ratios
Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are direction ratios, then a = λl, b = λm, c = λn for any nonzero λ ∈ R.
e.g. If a line has direction ratios 2, -1, -2, its direction cosines are 2/3, -1/3, -2/3.
Direction Cosines of a Line Joining Two Points
The direction cosines of the line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) are (x2 - x1)/PQ, (y2 - y1)/PQ, (z2 - z1)/PQ, where PQ = √[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²].
e.g. For points (-2, 4, -5) and (1, 2, 3), PQ = √77, and direction cosines are 3/√77, -2/√77, 8/√77.
Vector Equation of a Line (Point and Direction)
The vector equation of a line passing through a point with position vector a and parallel to a given vector b is r = a + λb, where λ is a real parameter.
e.g. Line through (5, 2, -4) parallel to 3i + 2j - 8k has equation r = (5i + 2j - 4k) + λ(3i + 2j - 8k).
Cartesian Equation of a Line (Point and Direction)
The Cartesian equation of a line passing through point (x1, y1, z1) with direction ratios a, b, c is (x - x1)/a = (y - y1)/b = (z - z1)/c.
e.g. Line through (5, 2, -4) with direction ratios 3, 2, -8 has equation (x - 5)/3 = (y - 2)/2 = (z + 4)/(-8).
Angle Between Two Lines (Vector Form)
If θ is the acute angle between lines r = a1 + λb1 and r = a2 + μb2, then cos θ = |b1 · b2| / (|b1| |b2|).
e.g. For lines with b1 = i + 2j + 2k and b2 = 3i + 2j + 6k, cos θ = (3+4+12)/(3×7) = 19/21, so θ = cos⁻¹(19/21).
Angle Between Two Lines (Cartesian Form)
If two lines have direction ratios a1, b1, c1 and a2, b2, c2, then cos θ = |a1a2 + b1b2 + c1c2| / (√(a1²+b1²+c1²) √(a2²+b2²+c2²)).
e.g. For lines with direction ratios 3,5,4 and 1,1,2, cos θ = (3+5+8)/(√50 × √6) = 16/(5√2 × √6) = 8√3/15.
Skew Lines
Skew lines are lines in space which are neither intersecting nor parallel. They are non-coplanar and lie in different planes.
e.g. In a room, the line GE (diagonal across the ceiling) and the line DB (from one corner of the ceiling diagonally down the wall) are skew lines.
Shortest Distance Between Two Skew Lines (Vector Form)
The shortest distance d between skew lines r = a1 + λb1 and r = a2 + μb2 is given by d = |(b1 × b2) · (a2 - a1)| / |b1 × b2|.
e.g. For lines r = (i + j) + λ(2i - j + k) and r = (2i + j - k) + μ(3i - 5j + 2k), d = |(-3i - 7j - k) · (i - k)| / √59 = | -3 + 0 + 1| / √59 = 2/√59.
Shortest Distance Between Two Skew Lines (Cartesian Form)
The shortest distance between lines (x-x1)/a1 = (y-y1)/b1 = (z-z1)/c1 and (x-x2)/a2 = (y-y2)/b2 = (z-z2)/c2 is given by the absolute value of the determinant of the matrix with rows (x2-x1, y2-y1, z2-z1), (a1, b1, c1), (a2, b2, c2) divided by √[(b1c2 - b2c1)² + (c1a2 - c2a1)² + (a1b2 - a2b1)²].
e.g. For lines (x+1)/7 = (y+1)/(-6) = (z+1)/1 and (x-3)/1 = (y-5)/(-2) = (z-7)/1, the shortest distance is computed using the determinant formula.
Distance Between Parallel Lines
The distance between parallel lines r = a1 + λb and r = a2 + μb is d = |b × (a2 - a1)| / |b|.
e.g. For lines r = (2i + j - 4k) + λ(2i + 3j + 6k) and r = (3i + 3j - 5k) + μ(2i + 3j + 6k), d = |(2i+3j+6k) × (i+2j-k)| / 7 = √293 / 7.
Condition for Perpendicular Lines
Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are perpendicular if a1a2 + b1b2 + c1c2 = 0.
e.g. Lines with direction ratios 1, -1, 2 and 3, 4, -2 are perpendicular because 1×3 + (-1)×4 + 2×(-2) = 3 - 4 - 4 = -5 ≠ 0, so they are not perpendicular.
Condition for Parallel Lines
Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are parallel if a1/a2 = b1/b2 = c1/c2.
e.g. Lines through (4,7,8) and (2,3,4) have direction ratios -2, -4, -4; line through (-1,-2,1) and (1,2,5) has direction ratios 2, 4, 4. Since -2/2 = -4/4 = -4/4 = -1, they are parallel.
Formulas & Equations (12)
l² + m² + n² = 1
Reads as: l squared plus m squared plus n squared equals 1
Variables: l, m, n are direction cosines of a line (dimensionless)
Use when: To verify a set of direction cosines or to find direction cosines from direction ratios.
⚠ Students forget the sum of squares equals 1 and treat direction ratios as direction cosines directly.
l = a / √(a² + b² + c²), m = b / √(a² + b² + c²), n = c / √(a² + b² + c²)
Reads as: l equals a over root a squared plus b squared plus c squared, m equals b over root a squared plus b squared plus c squared, n equals c over root a squared plus b squared plus c squared
Variables: a, b, c are direction ratios; l, m, n are direction cosines (dimensionless)
Use when: To convert direction ratios to direction cosines.
⚠ Students forget the ± sign and take only positive values, losing marks when the line is directed opposite.
Direction cosines of line joining P(x1,y1,z1) and Q(x2,y2,z2): (x2-x1)/PQ, (y2-y1)/PQ, (z2-z1)/PQ where PQ = √[(x2-x1)² + (y2-y1)² + (z2-z1)²]
Reads as: x2 minus x1 over PQ, y2 minus y1 over PQ, z2 minus z1 over PQ
Variables: x1,y1,z1 and x2,y2,z2 are coordinates of two points; PQ is the distance between them
Use when: To find direction cosines of a line segment given two endpoints.
⚠ Students use (x1-x2) instead of (x2-x1) or forget to divide by PQ.
Vector equation of line: r = a + λb
Reads as: r equals a plus lambda b
Variables: r is position vector of any point on line; a is position vector of a fixed point; b is a vector parallel to the line; λ is a real parameter
Use when: To represent a line in vector form given a point and direction.
⚠ Students confuse a and b, writing r = b + λa instead.
Cartesian equation of line: (x - x1)/a = (y - y1)/b = (z - z1)/c
Reads as: x minus x1 over a equals y minus y1 over b equals z minus z1 over c
Variables: (x1,y1,z1) is a fixed point; a,b,c are direction ratios; (x,y,z) is any point on the line
Use when: To write the equation of a line in Cartesian form given a point and direction ratios.
⚠ Students write (x - x1)/a = (y - y1)/b = (z - z1)/c but forget to handle negative signs in denominators correctly.
cos θ = |a1a2 + b1b2 + c1c2| / (√(a1²+b1²+c1²) √(a2²+b2²+c2²))
Reads as: cos theta equals absolute value of a1 a2 plus b1 b2 plus c1 c2 divided by root of a1 squared plus b1 squared plus c1 squared times root of a2 squared plus b2 squared plus c2 squared
Variables: a1,b1,c1 and a2,b2,c2 are direction ratios of two lines; θ is the acute angle between them
Use when: To find the angle between two lines given their direction ratios.
⚠ Students forget the absolute value and give an obtuse angle instead of the acute angle.
cos θ = |l1l2 + m1m2 + n1n2|
Reads as: cos theta equals absolute value of l1 l2 plus m1 m2 plus n1 n2
Variables: l1,m1,n1 and l2,m2,n2 are direction cosines of two lines; θ is the acute angle between them
Use when: To find the angle between two lines given their direction cosines.
⚠ Students forget the absolute value and write cos θ = l1l2 + m1m2 + n1n2, which can be negative.
sin θ = √[(l1m2 - l2m1)² + (m1n2 - m2n1)² + (n1l2 - n2l1)²]
Reads as: sin theta equals square root of l1 m2 minus l2 m1 squared plus m1 n2 minus m2 n1 squared plus n1 l2 minus n2 l1 squared
Variables: l1,m1,n1 and l2,m2,n2 are direction cosines of two lines; θ is the acute angle between them
Use when: To find sin of the angle between two lines, often used in problems involving perpendicularity or cross product.
⚠ Students confuse this with the formula for cos θ and misapply it.
Shortest distance between skew lines: d = |(b1 × b2) · (a2 - a1)| / |b1 × b2|
Reads as: d equals absolute value of b1 cross b2 dot a2 minus a1 divided by magnitude of b1 cross b2
Variables: a1, a2 are position vectors of points on lines; b1, b2 are direction vectors of lines; d is the shortest distance
Use when: To find the shortest distance between two skew lines given in vector form.
⚠ Students compute (b1 × b2) · (a2 - a1) but forget to take absolute value, leading to a negative distance.
Distance between parallel lines: d = |b × (a2 - a1)| / |b|
Reads as: d equals magnitude of b cross a2 minus a1 divided by magnitude of b
Variables: a1, a2 are position vectors of points on lines; b is the common direction vector; d is the distance
Use when: To find the distance between two parallel lines given in vector form.
⚠ Students use the skew line formula instead, which gives zero because b1 × b2 = 0 for parallel lines.
Condition for perpendicular lines: a1a2 + b1b2 + c1c2 = 0
Reads as: a1 a2 plus b1 b2 plus c1 c2 equals zero
Variables: a1,b1,c1 and a2,b2,c2 are direction ratios of two lines
Use when: To check if two lines are perpendicular.
⚠ Students set a1/a2 + b1/b2 + c1/c2 = 0 instead of the sum of products.
Condition for parallel lines: a1/a2 = b1/b2 = c1/c2
Reads as: a1 over a2 equals b1 over b2 equals c1 over c2
Variables: a1,b1,c1 and a2,b2,c2 are direction ratios of two lines
Use when: To check if two lines are parallel.
⚠ Students check only two ratios and ignore the third, or forget that the ratios must be equal, not just proportional.
How It Works
Finding Direction Cosines from Direction Ratios
  1. Step 1: Identify the direction ratios a, b, c of the line.
  2. Step 2: Compute the magnitude √(a² + b² + c²).
  3. Step 3: Divide each direction ratio by this magnitude: l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²).
  4. Step 4: Choose the appropriate sign (±) based on the direction of the line.
💡 The sum of squares of direction cosines must always equal 1. Use this to verify your answer.
Deriving Cartesian Equation of a Line from Vector Form
  1. Step 1: Write the vector equation r = a + λb, where a = x1i + y1j + z1k and b = ai + bj + ck.
  2. Step 2: Write r = xi + yj + zk and equate coefficients: x = x1 + λa, y = y1 + λb, z = z1 + λc.
  3. Step 3: Solve each for λ: λ = (x - x1)/a, λ = (y - y1)/b, λ = (z - z1)/c.
  4. Step 4: Equate the expressions: (x - x1)/a = (y - y1)/b = (z - z1)/c.
💡 If any direction ratio is zero, the corresponding numerator is also set to zero, not written as a fraction.
Finding Angle Between Two Lines
  1. Step 1: Identify the direction ratios (or direction cosines) of both lines.
  2. Step 2: Use the formula cos θ = |a1a2 + b1b2 + c1c2| / (√(a1²+b1²+c1²) √(a2²+b2²+c2²)) for direction ratios.
  3. Step 3: Compute the numerator and denominator separately.
  4. Step 4: Simplify to get cos θ, then find θ = cos⁻¹(value).
💡 Always take the absolute value to ensure θ is the acute angle. If the question asks for the angle between lines, it is always the acute angle.
Finding Shortest Distance Between Two Skew Lines (Vector Method)
  1. Step 1: Write the equations in the form r = a1 + λb1 and r = a2 + μb2.
  2. Step 2: Compute the cross product b1 × b2.
  3. Step 3: Compute the vector difference a2 - a1.
  4. Step 4: Compute the scalar triple product (b1 × b2) · (a2 - a1).
  5. Step 5: Take the absolute value of this scalar triple product.
  6. Step 6: Divide by the magnitude |b1 × b2| to get the shortest distance d.
💡 If b1 × b2 = 0, the lines are parallel, not skew. Use the parallel line distance formula instead.
Finding Distance Between Two Parallel Lines
  1. Step 1: Write the equations in the form r = a1 + λb and r = a2 + μb (same direction vector b).
  2. Step 2: Compute the vector difference a2 - a1.
  3. Step 3: Compute the cross product b × (a2 - a1).
  4. Step 4: Find the magnitude of this cross product.
  5. Step 5: Divide by the magnitude |b| to get the distance d.
💡 The cross product b × (a2 - a1) gives a vector perpendicular to both b and the line joining the two points. Its magnitude divided by |b| gives the perpendicular distance.
Checking Collinearity of Three Points
  1. Step 1: Find direction ratios of line joining first two points (e.g., A and B).
  2. Step 2: Find direction ratios of line joining second and third points (e.g., B and C).
  3. Step 3: Check if the two sets of direction ratios are proportional.
  4. Step 4: If proportional and the common point (B) lies on both lines, the points are collinear.
💡 Proportionality of direction ratios alone is not enough; the lines must share a common point. Always verify the common point.
Exam Traps — Don't Make These Mistakes
⚠ Confusing Direction Ratios with Direction Cosines
✗ Students write direction ratios as the cosines of angles, e.g., writing direction cosines as (2, -1, -2) instead of (2/3, -1/3, -2/3).
→ Direction ratios are proportional to direction cosines, not equal. Direction cosines must satisfy l² + m² + n² = 1, which (2, -1, -2) does not.
✓ First find the magnitude √(a²+b²+c²) = √(4+1+4) = 3, then divide: l = 2/3, m = -1/3, n = -2/3.
⚠ Forgetting the ± Sign in Direction Cosines
✗ Students write direction cosines as only positive values, e.g., (2/3, 1/3, 2/3) for direction ratios (2, -1, -2).
→ Direction cosines depend on the direction of the line. If the line is directed opposite, the signs reverse. The formula includes a ± sign.
✓ Write the direction cosines with the correct signs: (2/3, -1/3, -2/3) or (-2/3, 1/3, 2/3) depending on the direction.
⚠ Using Wrong Formula for Angle Between Lines
✗ Students use cos θ = (a1a2 + b1b2 + c1c2) / (a1²+b1²+c1² + a2²+b2²+c2²) or similar incorrect forms.
→ The correct denominator is the product of the magnitudes, not the sum of squares. The formula is cos θ = |a1a2 + b1b2 + c1c2| / (√(a1²+b1²+c1²) √(a2²+b2²+c2²)).
✓ Compute √(a1²+b1²+c1²) and √(a2²+b2²+c2²) separately, multiply them, and divide the dot product by this product.
⚠ Forgetting Absolute Value in Shortest Distance Formula
✗ Students write d = (b1 × b2) · (a2 - a1) / |b1 × b2| without absolute value in the numerator.
→ The scalar triple product can be negative, leading to a negative distance, which is meaningless. Distance is always non-negative.
✓ Always take the absolute value: d = |(b1 × b2) · (a2 - a1)| / |b1 × b2|.
⚠ Misidentifying Skew vs. Parallel Lines
✗ Students apply the skew line distance formula to parallel lines, getting a division by zero or an incorrect result.
→ For parallel lines, b1 × b2 = 0, so the skew line formula is undefined. Parallel lines require a different formula.
✓ First check if b1 and b2 are proportional. If yes, use the parallel line distance formula: d = |b × (a2 - a1)| / |b|.
⚠ Incorrectly Writing Equation of Line When a Direction Ratio is Zero
✗ Students write (x - x1)/0 = (y - y1)/b = (z - z1)/c, which is undefined.
→ Division by zero is not allowed. If a direction ratio is zero, the line is parallel to the corresponding axis, and the equation must reflect that.
✓ If a = 0, write x = x1, and then (y - y1)/b = (z - z1)/c. For example, a line parallel to the y-axis through (1,2,3) is x = 1, z = 3, and y varies.
Compare & Contrast
Definition and properties
Direction Cosines
Direction Ratios
Direction cosines are the actual cosines of the angles made with axes and satisfy l² + m² + n² = 1. Direction ratios are any three numbers proportional to direction cosines and do not necessarily satisfy the sum of squares condition.
Relative orientation and coplanarity
Skew Lines
Parallel Lines
Skew lines are non-coplanar, neither parallel nor intersecting, and have a unique shortest distance perpendicular to both. Parallel lines are coplanar, have the same direction, and the shortest distance is the perpendicular distance from a point on one line to the other.
Exam Focus
Marks pattern: This chapter typically appears with 1-2 MCQs (1 mark each), 1 short answer question (3 marks), and 1 long answer question (5 marks), totaling around 10-12 marks in the board exam.
Most asked: Finding direction cosines and direction ratios of a line · Writing vector and Cartesian equations of a line · Finding the angle between two lines · Finding the shortest distance between skew lines · Checking collinearity of three points · Conditions for perpendicular and parallel lines
Don't skip: Shortest distance between two skew lines (vector and Cartesian form) — 5 marks; Angle between two lines and related conditions — 3-5 marks; Equation of a line in space — 3 marks
One-Line Revision (read aloud before the exam)
  1. Direction cosines are the cosines of the angles a line makes with the positive x, y, and z axes, and satisfy l² + m² + n² = 1.
  2. Direction ratios are any three numbers proportional to direction cosines.
  3. To convert direction ratios (a,b,c) to direction cosines, divide each by √(a²+b²+c²) and include the appropriate ± sign.
  4. The direction cosines of a line joining P(x1,y1,z1) and Q(x2,y2,z2) are (x2-x1)/PQ, (y2-y1)/PQ, (z2-z1)/PQ.
  5. The vector equation of a line is r = a + λb, where a is a point on the line and b is a direction vector.
  6. The Cartesian equation of a line is (x - x1)/a = (y - y1)/b = (z - z1)/c.
  7. The angle between two lines is given by cos θ = |a1a2 + b1b2 + c1c2| / (√(a1²+b1²+c1²) √(a2²+b2²+c2²)).
  8. Two lines are perpendicular if a1a2 + b1b2 + c1c2 = 0.
  9. Two lines are parallel if a1/a2 = b1/b2 = c1/c2.
  10. Skew lines are non-coplanar, neither parallel nor intersecting.
  11. The shortest distance between skew lines is d = |(b1 × b2) · (a2 - a1)| / |b1 × b2|.
  12. The distance between parallel lines is d = |b × (a2 - a1)| / |b|.
  13. Three points A, B, C are collinear if direction ratios of AB and BC are proportional and B is common.
  14. If a direction ratio is zero, the line is parallel to the corresponding axis and the equation is written with the numerator set to zero.
  15. Always take the acute angle between lines by using the absolute value in the cos θ formula.

MCQs — 10 Questions with Answers

Q1. If a line makes angles 90°, 60°, and 30° with the positive x, y, and z axes respectively, what are its direction cosines?

  • A. 0, 1/2, √3/2 ✓
  • B. 1, 1/2, √3/2
  • C. 0, 1/√3, 1/√3
  • D. √3/2, 1/2, 0

Answer: A — Direction cosines are cos 90° = 0, cos 60° = 1/2, and cos 30° = √3/2.

Q2. A line has direction ratios 1, 2, 2. Which of the following are its direction cosines?

  • A. 1/3, 2/3, 2/3 ✓
  • B. 1/2, 2/2, 2/2
  • C. 1/9, 2/9, 2/9
  • D. 1/√5, 2/√5, 2/√5

Answer: A — √(1² + 2² + 2²) = √9 = 3, so direction cosines are 1/3, 2/3, 2/3, and 1/9 + 4/9 + 4/9 = 1 ✓

Q3. What are the direction cosines of the y-axis?

  • A. 1, 0, 0
  • B. 0, 1, 0 ✓
  • C. 0, 0, 1
  • D. 1, 1, 1

Answer: B — The y-axis makes angles 90°, 0°, 90° with x, y, z axes, giving direction cosines 0, 1, 0.

Q4. Find the direction ratios of the line passing through points A(1, 2, 3) and B(4, 5, 6).

  • A. 3, 3, 3 ✓
  • B. 1, 1, 1
  • C. 4, 5, 6
  • D. −3, −3, −3

Answer: A — Direction ratios = (4−1, 5−2, 6−3) = (3, 3, 3); note (B) is a simplified proportional form, but (A) is exact.

Q5. If l, m, n are direction cosines of a line, which statement is always true?

  • A. l + m + n = 1
  • B. l² + m² + n² = 1 ✓
  • C. l × m × n = 1
  • D. l − m − n = 0

Answer: B — The fundamental property of direction cosines is that the sum of their squares equals 1.

Q6. Show that points A(2, 3, −4), B(1, −2, 3), and C(3, 8, −11) are collinear. Which condition must be satisfied?

  • A. Direction ratios of AB and BC are equal
  • B. Direction ratios of AB and BC are proportional ✓
  • C. Distance AB = Distance BC
  • D. The sum of coordinates of A and C equals B

Answer: B — Collinearity requires proportional direction ratios: AB has ratios (−1, −5, 7) and BC has (2, 10, −14) = −2 × (−1, −5, 7).

Q7. Which of the following is NOT a correct statement about direction cosines and direction ratios?

  • A. Direction ratios are proportional to direction cosines
  • B. Direction cosines satisfy l² + m² + n² = 1, but direction ratios do not
  • C. A line has a unique set of direction cosines if treated as directed
  • D. Direction ratios and direction cosines have the same numerical values ✓

Answer: D — Direction cosines and ratios have the same values only by coincidence; in general, ratios must be normalized by √(a² + b² + c²) to get cosines.

Q8. If direction cosines of a line are l = 2/√29, m = 3/√29, n = ?. What is n?

  • A. √(29 − 4 − 9)/√29 = √16/√29 = 4/√29
  • B. −4/√29
  • C. ±4/√29 ✓
  • D. Cannot be determined

Answer: C — Using l² + m² + n² = 1: (4 + 9)/29 + n² = 1 gives n² = 16/29, so n = ±4/√29.

Q9. Two lines have direction ratios (1, 2, 3) and (2, 4, 6) respectively. What is their relationship?

  • A. They are perpendicular
  • B. They are parallel ✓
  • C. They intersect at 45°
  • D. They are skew lines

Answer: B — The second set (2, 4, 6) = 2 × (1, 2, 3), so direction ratios are proportional, meaning the lines are parallel.

Q10. The Cartesian equation of a line passing through (2, 3, 4) with direction ratios 1, −1, 2 is (x − 2)/1 = (y − 3)/(−1) = (z − 4)/2. If a point (x, y, z) lies on this line with parameter value λ = 3, find the coordinates of the point.

  • A. (5, 0, 10) ✓
  • B. (2, 3, 4)
  • C. (3, 2, 6)
  • D. (−1, 6, 2)

Answer: A — From the Cartesian form: x = 2 + λ(1) = 2 + 3 = 5, y = 3 + λ(−1) = 3 − 3 = 0, z = 4 + λ(2) = 4 + 6 = 10.

Flashcards

What is the fundamental property that all direction cosines must satisfy?

The sum of squares of direction cosines equals one: l² + m² + n² = 1.

How do you convert direction ratios a, b, c into direction cosines l, m, n?

Divide each ratio by the square root of the sum of their squares: l = a/√(a² + b² + c²), and similarly for m and n.

Find the direction cosines of the x-axis.

The x-axis makes angles 0°, 90°, 90° with x, y, z axes respectively, so its direction cosines are 1, 0, 0.

What are the direction ratios of the line joining points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)?

The direction ratios are x₂ − x₁, y₂ − y₁, z₂ − z₁ (or their negative multiples).

If a line passes through point A with position vector **a** and is parallel to vector **b**, what is its vector equation?

The vector equation is **r** = **a** + λ**b**, where λ is a real parameter.

Write the Cartesian equation of a line through point (x₁, y₁, z₁) with direction ratios a, b, c.

The Cartesian form is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.

How do you check if three points A, B, C are collinear using direction ratios?

Find direction ratios of AB and BC; if they are proportional, the points are collinear.

State the formula for the direction cosines of a line joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).

Direction cosines are (x₂ − x₁)/PQ, (y₂ − y₁)/PQ, (z₂ − z₁)/PQ, where PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].

Why does a given line in space have two sets of direction cosines?

A line can be extended in two opposite directions, and the direction cosines change sign when the direction reverses.

If direction ratios of a line are 2, −1, −2, what are its direction cosines?

First find √(4 + 1 + 4) = 3, then direction cosines are ±(2/3, −1/3, −2/3).

Important Board Questions

If a line makes equal angles with the coordinate axes, find its direction cosines. [2 marks]

Let each angle be θ; then l = m = n = cos θ. Use the constraint l² + m² + n² = 1 to solve for cos θ, giving l = m = n = ±1/√3.

Derive the formula for direction cosines of a line passing through two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). Hence, find the direction cosines of the line joining (−2, 4, −5) and (1, 2, 3). [5 marks]

The direction cosines are proportional to (x₂ − x₁, y₂ − y₁, z₂ − z₁); normalize by dividing by PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. Here PQ = √77, and direction cosines are 3/√77, −2/√77, 8/√77.

Using direction ratios, prove that the points A(3, 5, −4), B(−1, 1, 2), and C(−5, −5, −2) are collinear. Also, write the vector equation of the line passing through these points. [6 marks]

Calculate direction ratios of AB and BC; verify they are proportional (both should be multiples of (−4, −4, 6) or equivalent). Write vector equation as **r** = **a** + λ**b** where **a** is position vector of A and **b** is the common direction vector; alternatively verify using parametric form that all three points satisfy the same linear relationship.

Assertion-Reason Questions for Three Dimensional Geometry

Q1. Assertion (A):

The direction cosines of the x-axis are (1, 0, 0).

Reason (R):

The x-axis makes angles 0°, 90°, 90° with the x, y, and z axes respectively.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why the d.c.'s are the cosines of these angles: cos 0°=1, cos 90°=0, cos 90°=0.

Q2. Assertion (A):

If a line has direction ratios 3, 4, 5, then its direction cosines are 3/√50, 4/√50, 5/√50.

Reason (R):

Direction cosines are obtained by dividing direction ratios by the square root of the sum of their squares, taking positive values.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true (3/√50, 4/√50, 5/√50 are correct d.c.'s), and the reason is also true, but the reason does not uniquely explain why we take positive values—for an undirected line, negative values are equally valid. The reason is incomplete.

Q3. Assertion (A):

Two distinct lines in space can have the same set of direction cosines.

Reason (R):

Direction cosines depend only on the slope of a line, not on its position.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true. Parallel lines (or differently positioned lines with the same slope) share the same direction cosines. The reason correctly explains the assertion.

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Case-Based Questions (CBQ) for Three Dimensional Geometry

Case Study:

An architectural firm is designing a complex structure with three support beams anchored at different points in 3D space. Beam 1 originates from point A(0, 0, 0) and extends toward point B(3, 4, 0). Beam 2 runs from point C(1, 2, 3) toward point D(4, 6, 9). Beam 3 connects points E(2, 1, 5) and F(5, 5, 5). The engineers need to determine the direction and slope of each beam to ensure structural stability and load distribution. They use direction cosines to characterize the orientation of each beam in 3D space.

Q1. What are the direction ratios of Beam 1 from A to B?

  • A. (3, 4, 0) ✓
  • B. (0, 0, 0)
  • C. (3, 4, 1)
  • D. (1, 1, 0)

Answer: A

Q2. Find the direction cosines of Beam 2 from C to D.

  • A. (3/√54, 4/√54, 6/√54)
  • B. (1/√3, 1/√3, 1/√3) ✓
  • C. (3/√27, 4/√27, 6/√27)
  • D. (1/√27, 2/√27, 2/√27)

Answer: B

Q3. What is the relationship between Beam 1 and Beam 2 in terms of their directions?

  • A. They are parallel lines
  • B. They have proportional direction ratios
  • C. They are skew lines with different directions ✓
  • D. They intersect at a right angle

Answer: C

Q4. If Beam 3 has direction cosines (l, m, n) with l = 3/5 and m = 4/5, what is the value of n?

  • A. ±1/5
  • B. 0 ✓
  • C. ±3/5
  • D. Cannot be determined

Answer: B

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Previous Year Questions (PYQ) for Three Dimensional Geometry

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 65/1/1 · 1 mark

The length of perpendicular drawn from point (2, 5, 7) on line x/1 = y/0 = z/0 is

  • 2
  • 5
  • √74
  • √78

CBSE 2026 · Set 65/1/1 · 1 mark

Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other when pp' + rr' = 1. Reason (R): Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are perpendicular to each other if b₁ · b₂ = 0.

  • Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
  • Assertion (A) is true and Reason (R) is false.
  • Assertion (A) is false and Reason (R) is true.

CBSE 2026 · Set 65/1/1 · 5 marks

Check whether the lines given by (x – 1)/2 = (y – 2)/3 = (z – 3)/4 and (x – 4)/5 = (y – 1)/2 = z are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

CBSE 2026 · Set 65/1/2 · 1 mark

The length of perpendicular drawn from the point (1, 2, 3) on line x/0 = y/1 = z/0 is

  • 2
  • 6
  • 10
  • 14

CBSE 2026 · Set 65/1/2 · 5 marks

Prove that the line through points A(0, –1, –1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(–4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.

CBSE 2026 · Set 65/1/3 · 1 mark

The length of perpendicular drawn from the point (3, 4, 2) on the line x/0 = y/0 = z/1 is

  • 2
  • 9
  • 5
  • √29

CBSE 2026 · Set 65/1/3 · 1 mark

The value of m for which the points with position vectors -i - j + 2k, 2i + mj + 5k and 3i + 11j + 6k are collinear, is

  • 8
  • -8
  • 2
  • 5/2

CBSE 2026 · Set 65/1/3 · 5 marks

Show that line AB passing through points A(0, 4, 1), B(2, 3, –1) and the line CD passing through points C(4, 5, 0), D(2, 6, 2) are parallel. Also, find distance between them.

CBSE 2026 · Set 65/2/1 · 1 mark

Direction cosines of the line given by equations: (2x - 1)/4 = (y - 1)/3 = (z - 6)/6 are

  • 2, -3, -6
  • -6/7, -3/7, 2/7
  • 6/7, -3/7, 2/7
  • -6/√61, -3/√61, 4/√61

CBSE 2026 · Set 65/2/1 · 1 mark

Assertion (A): A line can have direction cosines < 1, 1, 1 >. Reason (R): cos α = 1 is possible for α = 0.

  • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
  • Assertion (A) is true, but Reason (R) is false.
  • Assertion (A) is false, but Reason (R) is true.

CBSE 2026 · Set 65/2/1 · 2 marks

Find the co-ordinates of the point on the line r⃗ = -ĵ + 3k̂ + λ(2î - 2ĵ + k̂) such that the sum of co-ordinates is 3.

CBSE 2026 · Set 65/2/1 · 5 marks

A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line r⃗ = 4î + ĵ + λ(5î + 2ĵ + k̂). Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

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Class 12 Maths Formula Sheet

Every formula from all 13 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 12 Mathematics Previous Year Questions, Chapter-wise

All 1048 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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