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State Board · Class 12 · Mathematics Part 1

Lesson 6: Line and Plane

4 concepts · 32 practice questions with answers · Free to read

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Equation of a line in space

From daily life

A taut wire passes through a given point in a given direction; point and direction fix the line completely. Building on class 11 : the plane line y = mx + c now moves into three dimensions.

Simple definition

The line through the point A (position vector a) in the direction of b has the vector form r = a + λb and the Cartesian form (x − x₁)/l = (y − y₁)/m = (z − z₁)/n, where (l, m, n) are direction ratios.

Rule 1: Point and direction — r = a + λb.

Rule 2: Two points — Direction = b − a; then r = a + λ(b − a).

Abr = a + λba

In the figure : a fixes the point and b the direction.

How is the formula obtained? : A(1, 2, 3), b = (2, −1, 2): x = 1 + 2λ, y = 2 − λ, z = 3 + 2λ, giving (x − 1)/2 = (y − 2)/(−1) = (z − 3)/2.

  • Easy : Through the origin along (1, 1, 1): r = λ(1, 1, 1).
  • Medium : A(1, 2, 3), B(4, 5, 9): direction (3, 3, 6) ~ (1, 1, 2), so (x − 1)/1 = (y − 2)/1 = (z − 3)/2.
  • Exam-type : First line above and plane x + y + z = 12: (1 + 2λ) + (2 − λ) + (3 + 2λ) = 6 + 3λ = 12 ⇒ λ = 2, point (5, 0, 7).
Remember

Multiplying the direction ratios by a number gives the same line.

Common mistake

Writing the wrong sign of the point's coordinates in the Cartesian form.

Shortcut

Given two points, subtract; that is the direction. Memory clue : "Point + direction = line".

CET trap

If a denominator is 0 (say n = 0), write z − z₁ = 0 separately.

CET-style example : The line through (1, 2, 3) with direction (2, −1, 2) is: (A) (x−1)/2 = (y−2)/(−1) = (z−3)/2 (B) (x+1)/2 = (y+2)/(−1) = (z+3)/2 (C) (x−2)/1 = (y+1)/2 = (z−2)/3 (D) x/1 = y/2 = z/3

Answer : (A). Time-saving method : Point subtracted on top, direction below.

Spot the difference

वापरUse
r = a + λbr = a + λb
(x − x₁)/l = ...(x − x₁)/l = ...
दिशा = वजाबाकीdirection = difference

Practice questions

Easy
  1. 1Write the line through the origin along (2, 3, 6).
  2. 2Vector line through (1, 2, 3) along (2, −1, 2)?
  3. 3Cartesian form of the same line?
Medium
  1. 4Line through A(1, 2, 3) and B(4, 5, 9)?
  2. 5The point of the first line for λ = 2?
  3. 6Where does r = (1, 2, 3) + λ(2, −1, 2) meet the plane x + y + z = 12?
Challenging
  1. 7Check that (5, 0, 7) lies on the plane and on the line.
  2. 8Direction cosines of a line with ratios (2, 1, −2)?
Show answers (try them yourself first)
  1. 1x/2 = y/3 = z/6.
  2. 2r = (1, 2, 3) + λ(2, −1, 2).
  3. 3(x − 1)/2 = (y − 2)/(−1) = (z − 3)/2.
  4. 4(x − 1)/1 = (y − 2)/1 = (z − 3)/2.
  5. 5(5, 0, 7).
  6. 6(5, 0, 7).
  7. 75 + 0 + 7 = 12; on the line with λ = 2.
  8. 8√9 = 3, so (2/3, 1/3, −2/3).

Angle between lines and shortest distance

From daily life

The road under a flyover and the road on it never meet, yet their smallest gap can be measured; these are skew lines.

Simple definition

The angle between lines with direction vectors b₁, b₂ is given by cos θ = |b₁·b₂| / (|b₁||b₂|). Skew lines neither meet nor are parallel. Their shortest distance is the length of the segment perpendicular to both.

Rule 1: Skew lines — SD = |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|.

Rule 2: Parallel lines — Distance = |b × (a₂ − a₁)| / |b|.

Rule 3: Intersecting lines — SD = 0, i.e. (a₂ − a₁)·(b₁ × b₂) = 0.

L₁L₂SD

In the figure : The common perpendicular joining L₁ and L₂ gives SD.

How is the formula obtained? : X-axis (a₁ = 0, b₁ = (1, 0, 0)) and the line through (0, 0, 3) along (0, 1, 0): b₁ × b₂ = (0, 0, 1), (a₂ − a₁)·(0, 0, 1) = 3, |b₁ × b₂| = 1, so SD = 3.

  • Easy : b₁ = (1, 2, 2), b₂ = (2, −1, 2): cos θ = 4/9.
  • Medium : X-axis and the Y-direction line through (0, 0, 3): SD = 3.
  • Exam-type : a₁ = (1, 2, 3), b₁ = (1, 1, 1), a₂ = (2, 4, 5), b₂ = (2, 1, 0): b₁ × b₂ = (−1, 2, −1), (a₂ − a₁)·n = −1 + 4 − 2 = 1, |n| = √6, so SD = 1/√6 ≈ 0.41.
Remember

If the lines are parallel b₁ × b₂ = 0, so use the parallel formula instead.

Common mistake

Using (a₁ − a₂) flips the sign; forgetting the modulus.

Shortcut

Numerator is a triple product, denominator is the magnitude of the cross product. Memory clue : "Triple product ÷ cross = SD".

CET trap

Do not use the skew formula for parallel lines (0/0).

CET-style example : The shortest distance between the X-axis and the Y-direction line through (0, 0, 3) is: (A) 3 (B) 0 (C) 1 (D) √3

Answer : (A). Time-saving method : Picture it: the z-gap is 3.

Spot the difference

अंतरDistance
(a₂−a₁)·(b₁×b₂)/b₁×b₂(a₂−a₁)·(b₁×b₂)/b₁×b₂
b×(a₂−a₁)/bb×(a₂−a₁)/b
00

Practice questions

Easy
  1. 1Angle between lines with directions (1, 0, 0) and (0, 1, 0)?
  2. 2cos θ for lines along (1, 2, 2) and (2, −1, 2)?
  3. 3Are the lines along (1, 1, 1) and (1, −2, 1) perpendicular?
Medium
  1. 4X-axis and the Y-direction line through (0, 0, 3): SD?
  2. 5For b₁ = (1, 1, 1), b₂ = (2, 1, 0) find b₁ × b₂.
  3. 6a₁ = (1, 2, 3), a₂ = (2, 4, 5) with the above directions: SD?
Challenging
  1. 7Distance between r = λ(1, 2, 2) and r = (1, 0, 0) + μ(1, 2, 2)?
  2. 8If (a₂ − a₁)·(b₁ × b₂) = 0, what can be said of the lines?
Show answers (try them yourself first)
  1. 190°.
  2. 24/9.
  3. 3Yes, the dot product is 0.
  4. 43.
  5. 5(−1, 2, −1).
  6. 61/√6 ≈ 0.41.
  7. 7|b × (a₂ − a₁)|/|b| = √8/3 = 2√2/3 ≈ 0.94.
  8. 8They lie in one plane: they intersect (or are parallel).

Equation of a plane

From daily life

A tabletop is fixed by one point on it and the direction perpendicular to it, like a nail driven straight in.

Simple definition

A plane is a flat surface in space. The plane through P₀(x₀, y₀, z₀) with normal n = (a, b, c) has equation a(x − x₀) + b(y − y₀) + c(z − z₀) = 0, i.e. ax + by + cz = d.

Rule 1: Point and normal — n·(r − a) = 0.

Rule 2: Intercept form — x/p + y/q + z/r = 1 (intercepts p, q, r on the axes).

Rule 3: Three points — n = AB × AC, then use the point-normal form.

nप्रतल

In the figure : The normal n is perpendicular to every line in the plane.

How is the formula obtained? : Through (1, 2, 3) with n = (2, −1, 2): 2(x − 1) − (y − 2) + 2(z − 3) = 0 ⇒ 2x − y + 2z = 6.

  • Easy : 2x + 3y + 6z = 12 ⇒ x/6 + y/4 + z/2 = 1; intercepts 6, 4, 2.
  • Medium : A(1, 0, 0), B(0, 2, 0), C(0, 0, 3): AB × AC = (6, 3, 2), so 6x + 3y + 2z = 6, i.e. x/1 + y/2 + z/3 = 1.
  • Exam-type : Normal form of 2x + 3y + 6z = 12: n̂ = (2, 3, 6)/7, p = 12/7, i.e. (2/7)x + (3/7)y + (6/7)z = 12/7.
Remember

The coefficients of x, y, z are the components of the normal.

Common mistake

Forgetting to bring the constant to 1 in the intercept form.

Shortcut

For three points, AB × AC is directly the normal. Memory clue : "Coefficients = normal".

CET trap

Parallel planes share the normal; only the constant differs.

CET-style example : The z-intercept of 2x + 3y + 6z = 12 is: (A) 2 (B) 6 (C) 4 (D) 12

Answer : (A). x = y = 0 gives z = 2. Time-saving method : Set the other two coordinates to 0.

Spot the difference

नमुनाPattern
ax + by + cz = dax + by + cz = d
x/p + y/q + z/r = 1x/p + y/q + z/r = 1
r·n = dr·n = d

Practice questions

Easy
  1. 1Plane through the origin with normal (1, 1, 1)?
  2. 2Plane through (1, 2, 3) with n = (2, −1, 2)?
  3. 3x-intercept of 2x + 3y + 6z = 12?
Medium
  1. 4Write it in intercept form.
  2. 5Plane through A(1, 0, 0), B(0, 2, 0), C(0, 0, 3)?
  3. 6Does (1, 1, 1) lie on 2x + 3y + 6z = 12?
Challenging
  1. 7Distance of 2x + 3y + 6z = 12 from the origin?
  2. 8Plane parallel to 2x + 3y + 6z = k through (1, 1, 1)?
Show answers (try them yourself first)
  1. 1x + y + z = 0.
  2. 22x − y + 2z = 6.
  3. 36.
  4. 4x/6 + y/4 + z/2 = 1.
  5. 56x + 3y + 2z = 6.
  6. 62 + 3 + 6 = 11 ≠ 12, so no.
  7. 712/√(4 + 9 + 36) = 12/7.
  8. 82x + 3y + 6z = 11.

Angle between planes, line-plane angle and distance of a point

From daily life

The angle at which two walls meet follows from the angle between their normals; the angle a ladder makes with a wall follows from its direction and the normal.

Simple definition

The angle between planes equals the angle between their normals: cos θ = |n₁·n₂| / (|n₁||n₂|). The angle φ between a line (direction b) and a plane (normal n) is sin φ = |b·n| / (|b||n|). The distance of the point (x₁, y₁, z₁) from ax + by + cz = d is |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²).

Rule 1: Perpendicular / parallel planes — n₁·n₂ = 0 means perpendicular; n₁ = kn₂ means parallel.

Rule 2: Line and plane — b·n = 0 means the line is parallel to the plane; b = kn means perpendicular.

How is the formula obtained? : x + 2y + 2z = 1 and 2x − y + 2z = 3: n₁·n₂ = 2 − 2 + 4 = 4; |n₁| = |n₂| = 3, so cos θ = 4/9.

  • Easy : Distance of (1, 1, 1) from 2x + 3y + 6z = 12: |11 − 12|/7 = 1/7.
  • Medium : Line direction (1, 2, 2), plane normal (2, 3, 6): sin φ = (2 + 6 + 12)/(3·7) = 20/21.
  • Exam-type : Distance of (5, 0, 7) from 2x − y + 2z = 6: |10 + 14 − 6|/3 = 6.
Remember

For a line and a plane use sin, not cos.

Common mistake

Using cos for the line-plane angle.

Shortcut

In the distance formula substitute the point, take the modulus and divide by the normal's magnitude. Memory clue : "Plane-plane = cos, line-plane = sin".

CET trap

Distance between parallel planes is |d₁ − d₂|/|n| (same n).

CET-style example : Distance of (1, 1, 1) from 2x + 3y + 6z = 12: (A) 1/7 (B) 1 (C) 7 (D) 11/7

Answer : (A). Time-saving method : Substitute and divide by 7.

Spot the difference

सूत्रFormula
cos θ =n₁·n₂/(n₁n₂)cos θ =n₁·n₂/(n₁n₂)
sin φ =b·n/(bn)sin φ =b·n/(bn)
cos θ =b₁·b₂/(b₁b₂)cos θ =b₁·b₂/(b₁b₂)

Practice questions

Easy
  1. 1Angle between x + y + z = 0 and x − y = 0?
  2. 2cos θ for x + 2y + 2z = 1 and 2x − y + 2z = 3?
  3. 3Distance of (1, 1, 1) from 2x + 3y + 6z = 12?
Medium
  1. 4Distance of the origin from 3x + 4y + 12z = 26?
  2. 5Line along (1, 2, 2) and plane 2x + 3y + 6z = 12: sin φ?
  3. 6Distance between 2x + 3y + 6z = 12 and 2x + 3y + 6z = 26?
Challenging
  1. 7Distance of (5, 0, 7) from 2x − y + 2z = 6?
  2. 8For what k is the line along (1, 2, k) parallel to x + y + z = 5?
Show answers (try them yourself first)
  1. 1n₁·n₂ = 0, so 90°.
  2. 24/9.
  3. 31/7.
  4. 426/13 = 2.
  5. 520/21.
  6. 614/7 = 2.
  7. 76.
  8. 81 + 2 + k = 0 ⇒ k = −3.