Is it correct to say: Reading the direction ratio of (1 − x)/3 as 3?

Rewrite (1 − x)/3 as (x − 1)/(−3), so the direction ratio is −3. Similarly, (2z − 1)/6 becomes (z − 1/2)/3.

Is it correct to say: Using a₁ and a₂ (the points) in the angle formula?

The angle between lines depends only on their directions. Use b₁ and b₂ alone.

Is it correct to say: Applying the skew-line formula to parallel lines?

For parallel lines b₁ × b₂ = 0, so that formula divides by zero. Use d = |b × (a₂ − a₁)|/|b|.

Is it correct to say: Giving a negative shortest distance or an obtuse angle?

Distance is a magnitude, so take the modulus. The angle between lines is taken as acute, so use |cos θ|.

Is it correct to say: Calling direction ratios direction cosines?

Direction cosines must satisfy l² + m² + n² = 1. If they do not, divide by √(a² + b² + c²).

Is it correct to say: Sign errors in the middle term of the cross product?

In the determinant expansion the ĵ term carries a minus sign: (b₁c₂ − b₂c₁)î − (a₁c₂ − a₂c₁)ĵ + (a₁b₂ − a₂b₁)k̂.

📄 Formula sheet (PDF)

Three Dimensional Geometry

Direction cosines and direction ratios, vector and Cartesian equations of a line, angle between two lines, and shortest distance between skew and parallel lines — NCERT Class 12 Maths Ch 11

📐 17 formulas✏️ 4 examples🎯 6 practice⚖️ Unit IV: 14 marks (with Ch 10)🏫 CBSE📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→Before reading direction ratios, bring each line to the standard form (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. A term like (1 − x)/3 means the direction ratio is −3.
→Check direction cosines with l² + m² + n² = 1. If the sum is not 1, you forgot to divide by √(a² + b² + c²).
→For the angle between two lines, use only the direction vectors b₁ and b₂, not the points a₁ and a₂.
→For shortest distance, first check whether the lines are parallel (direction ratios proportional). Parallel lines need a different formula, because b₁ × b₂ = 0.
→Lay out the cross-product determinant neatly. Arithmetic slips in the cross product are easy to make, so check each component.
→If the shortest distance comes out as 0, the lines intersect. Say so in your answer.

📐 Formulas(17)

Direction Cosines

l = cos α, m = cos β, n = cos γ

Symbol	Meaning
$α, β, γ$	Angles with the positive x, y and z-axes
$l, m, n$	Direction cosines (pure numbers)

Identity for Direction Cosines

l^{2} + m^{2} + n^{2} = 1

Direction Cosines from Direction Ratios★ Board fav

l = \pm \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \pm \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \pm \frac{c}{a ^{2} + b ^{2} + c ^{2}}

Symbol	Meaning
$a, b, c$	Direction ratios of the line

Direction Ratios of a Line Through Two Points

x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}

Direction Cosines of a Line Through Two Points

\frac{x _{2} - x _{1}}{P Q}, \frac{y _{2} - y _{1}}{P Q}, \frac{z _{2} - z _{1}}{P Q}, P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}

Symbol	Meaning
$P Q$	Distance between the two points (units)

Collinearity of Three Points

A, B, C collinear \Leftrightarrow d.r.s of A B and B C are proportional

Vector Equation of a Line

r = a + λ b

Symbol	Meaning
$a$	Position vector of a known point on the line
$b$	Vector parallel to the line (its components are direction ratios)
$λ$	Real parameter

Cartesian Equation of a Line★ Board fav

\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}

General Point on a Line

(x_{1} + λa, y_{1} + λb, z_{1} + λ c)

Line Through Two Points

r = a + λ (b - a), \frac{x - x _{1}}{x _{2} - x _{1}} = \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{z - z _{1}}{z _{2} - z _{1}}

Angle Between Two Lines (Vector Form)

cos θ = \frac{∣ b _{1} \cdot b _{2} ∣}{∣ b _{1} ∣ ∣ b _{2} ∣}

Angle Between Two Lines (Direction Ratios)★ Board fav

cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}

Angle Between Two Lines (Direction Cosines)

cos θ = ∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣

Perpendicular and Parallel Lines★ Board fav

⊥: a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0; ∥: \frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

Shortest Distance Between Skew Lines (Vector Form)★ Board fav

d = \frac{( b _{1} \times b _{2} ) \cdot ( a _{2} - a _{1} )}{∣ b _{1} \times b _{2} ∣}

Symbol	Meaning
$a_{1}, a_{2}$	Position vectors of known points on the two lines
$b_{1}, b_{2}$	Direction vectors of the two lines

Shortest Distance (Cartesian Form)

d = \frac{x _{2} - x _{1} a _{1} a _{2} y _{2} - y _{1} b _{1} b _{2} z _{2} - z _{1} c _{1} c _{2}}{( b _{1} c _{2} - b _{2} c _{1} ) ^{2} + ( c _{1} a _{2} - c _{2} a _{1} ) ^{2} + ( a _{1} b _{2} - a _{2} b _{1} ) ^{2}}

Distance Between Parallel Lines★ Board fav

d = \frac{∣ b \times ( a _{2} - a _{1} ) ∣}{∣ b ∣}

✏️ Solved Examples

1Solved Exampleeasy3 steps

A line has direction ratios 2, −3, 6. Find its direction cosines.

Find √(a² + b² + c²).

2^{2} + (- 3)^{2} + 6^{2} = 49 = 7

2Solved Exampleboard3 steps

Find the angle between the lines r = (3i + j − k) + λ(2i + 2j + k) and r = (i − 2j + 4k) + μ(4i + j + 8k).

Only the direction vectors matter.

b_{1} = 2 i^+ 2 j^+ k^, b_{2} = 4 i^+ j^+ 8 k^

3Solved Exampleboard4 steps

Find the shortest distance between the lines r = (i + 2j + 3k) + λ(i + 2j + 2k) and r = (2i − j + k) + μ(i + j).

Read off the points and direction vectors. The direction ratios 1, 2, 2 and 1, 1, 0 are not proportional, so the lines are not parallel.

a_{1} = i^+ 2 j^+ 3 k^, b_{1} = i^+ 2 j^+ 2 k^; a_{2} = 2 i^- j^+ k^, b_{2} = i^+ j^

4Solved ExampleHOTS4 steps

Find the distance between the lines r = (i + j) + λ(2i − j + 2k) and r = (2i + j − k) + μ(4i − 2j + 4k).

The direction vectors are 2i − j + 2k and 4i − 2j + 4k = 2(2i − j + 2k). They are proportional, so the lines are PARALLEL and the skew-line formula fails (b₁ × b₂ = 0).

b = 2 i^- j^+ 2 k^, ∣ b ∣ = 3

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Reading the direction ratio of (1 − x)/3 as 3
✓Rewrite (1 − x)/3 as (x − 1)/(−3), so the direction ratio is −3. Similarly, (2z − 1)/6 becomes (z − 1/2)/3.
2
Using a₁ and a₂ (the points) in the angle formula
✓The angle between lines depends only on their directions. Use b₁ and b₂ alone.
3
Applying the skew-line formula to parallel lines
✓For parallel lines b₁ × b₂ = 0, so that formula divides by zero. Use d = |b × (a₂ − a₁)|/|b|.
4
Giving a negative shortest distance or an obtuse angle
✓Distance is a magnitude, so take the modulus. The angle between lines is taken as acute, so use |cos θ|.
5
Calling direction ratios direction cosines
✓Direction cosines must satisfy l² + m² + n² = 1. If they do not, divide by √(a² + b² + c²).
6
Sign errors in the middle term of the cross product
✓In the determinant expansion the ĵ term carries a minus sign: (b₁c₂ − b₂c₁)î − (a₁c₂ − a₂c₁)ĵ + (a₁b₂ − a₂b₁)k̂.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the direction cosines of the line joining (2, −1, 3) and (4, 1, 4).
Q2
Find the vector and Cartesian equations of the line through (3, 0, −2) parallel to 2i − j + 5k.
Q3
Find k if the lines (x − 1)/2 = (y + 1)/k = (z − 2)/3 and (x + 3)/1 = (y − 2)/2 = (z + 1)/(−4) are perpendicular.
Q4
Write the line (2 − x)/3 = (y + 1)/4 = (2z − 1)/6 in vector form.
Q5
Find the shortest distance between the lines (x − 1)/1 = y/2 = (z + 1)/2 and (x − 2)/2 = (y − 1)/1 = (z − 3)/(−2).
Q6
Show that A(1, 2, 3), B(3, 5, 6) and C(7, 11, 12) are collinear.

📝 Notes

Three Dimensional Geometry

Chapter 11 describes a line in space by a point on it and a direction. Every formula here uses those two pieces: direction cosines or direction ratios for the direction, and a position vector or coordinates for the point.

Direction first

Direction cosines (l, m, n) are the cosines of the angles with the axes, and l² + m² + n² = 1.
Direction ratios (a, b, c) are any multiples of l, m, n. To get the cosines, divide by √(a² + b² + c²).
For the line through two points, the direction ratios are the differences of the coordinates.

Equations of a line

The vector form r = a + λb and the Cartesian form (x − x₁)/a = (y − y₁)/b = (z − z₁)/c describe the same line. Moving between them is a common short question: the point gives a, and the denominators give b. Always rewrite the equation so that x, y and z have coefficient +1 before reading anything off.

Angle and distance between lines

Angle: use only the direction vectors, cos θ = |b₁ · b₂|/(|b₁||b₂|). The lines are perpendicular when the dot product is 0 and parallel when the direction ratios are proportional.
Shortest distance: first check for parallel lines.
- Not parallel: d = |(b₁ × b₂) · (a₂ − a₁)|/|b₁ × b₂|. If d = 0, the lines intersect. Otherwise they are skew.
- Parallel: d = |b × (a₂ − a₁)|/|b|.

Scope reminder

In the rationalised NCERT textbook this chapter covers lines only. Equations of planes, and angles and distances involving planes, are not part of the current chapter.