📐 CHAPTER 11: THREE DIMENSIONAL GEOMETRY
Subject: Mathematics | Class: XII | CBSE
📐 DIRECTION COSINES AND DIRECTION RATIOS
🔹 Direction Cosines
Direction Cosines
If a directed line OP makes angles α, β, γ with positive X, Y, Z axes respectively, then:
l = cos α, m = cos β, n = cos γ
These are called direction cosines.
l² + m² + n² = 1
💡 Note: Direction cosines of a directed line are unique.
🔹 Direction Ratios
Direction Ratios
Numbers proportional to the direction cosines of a line are called direction ratios.
l/a = m/b = n/c
If a, b, c are direction ratios, then direction cosines are:
l = a/√(a²+b²+c²)
m = b/√(a²+b²+c²)
n = c/√(a²+b²+c²)
Direction ratios of line joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
a = x₂ - x₁, b = y₂ - y₁, c = z₂ - z₁
💡 Notes:
- Direction ratios of two parallel lines are proportional.
- Direction ratios of a line are not unique.
📝 EQUATION OF A STRAIGHT LINE
🔹 Line through a Given Point and Parallel to a Given Vector
Vector Form:
\vec{r} = \vec{a} + λ\vec{b}
where \vec{a} = position vector of point, \vec{b} = vector parallel to line
Cartesian Form:
(x - x₁)/a = (y - y₁)/b = (z - z₁)/c
where (x₁, y₁, z₁) is the point and a, b, c are direction ratios.
🔹 Line through Two Given Points
Vector Form:
\vec{r} = \vec{a} + λ(\vec{b} - \vec{a})
Cartesian Form:
(x - x₁)/(x₂ - x₁) = (y - y₁)/(y₂ - y₁) = (z - z₁)/(z₂ - z₁)
📐 ANGLE BETWEEN TWO LINES
🔹 Vector Form
For lines \vec{r} = \vec{a₁} + λ\vec{b₁} and \vec{r} = \vec{a₂} + μ\vec{b₂}
cos θ = (\vec{b₁} · \vec{b₂})/(|\vec{b₁}| |\vec{b₂}|)
🔹 Cartesian Form
For lines with DR's (a₁, b₁, c₁) and (a₂, b₂, c₂):
cos θ = (a₁a₂ + b₁b₂ + c₁c₂)/(√(a₁²+b₁²+c₁²) · √(a₂²+b₂²+c₂²))
🔹 Conditions
Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂
📏 SHORTEST DISTANCE BETWEEN TWO LINES
🔹 Skew Lines
Skew Lines
Two non-parallel and non-intersecting lines are called skew lines.
🔹 Vector Form
For lines \vec{r} = \vec{a₁} + λ\vec{b₁} and \vec{r} = \vec{a₂} + μ\vec{b₂}
SD = |(\vec{a₂} - \vec{a₁}) · (\vec{b₁} × \vec{b₂})| / |\vec{b₁} × \vec{b₂}|
🔹 Distance between Parallel Lines
For lines \vec{r} = \vec{a₁} + λ\vec{b} and \vec{r} = \vec{a₂} + μ\vec{b}
d = |(\vec{a₂} - \vec{a₁}) × \vec{b}| / |\vec{b}|
📐 PLANES
Plane
A surface such that a line segment joining any two points of it lies wholly on it.
A line perpendicular to every line lying on a plane is called a normal to the plane.
🔹 Equation of Plane in Normal Form
Vector Form: \vec{r} · \vec{n} = d
Cartesian Form: ax + by + cz = d
where a, b, c are direction ratios of normal.
🔹 Plane through a Point with Normal Vector
Vector Form: (\vec{r} - \vec{a}) · \vec{n} = 0
Cartesian Form: a(x - x₁) + b(y - y₁) + c(z - z₁) = 0
🔹 Plane through Three Non-Collinear Points
Vector Form:
(\vec{r} - \vec{a}) · [(\vec{b} - \vec{a}) × (\vec{c} - \vec{a})] = 0
🔹 Plane in Intercept Form
x/a + y/b + z/c = 1
where a, b, c are x, y, z intercepts respectively.
📐 PLANE PASSING THROUGH LINE OF INTERSECTION
Vector Form:
If planes are \vec{r}·\vec{n₁} = d₁ and \vec{r}·\vec{n₂} = d₂
\vec{r}·(\vec{n₁} + λ\vec{n₂}) = d₁ + λd₂
Cartesian Form:
If planes are a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂
(a₁x + b₁y + c₁z - d₁) + λ(a₂x + b₂y + c₂z - d₂) = 0
📐 ANGLE BETWEEN TWO PLANES
🔹 Vector Form
For planes \vec{r}·\vec{n₁} = d₁ and \vec{r}·\vec{n₂} = d₂
cos θ = (\vec{n₁} · \vec{n₂})/(|\vec{n₁}| |\vec{n₂}|)
🔹 Cartesian Form
For planes a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂
cos θ = (a₁a₂ + b₁b₂ + c₁c₂)/(√(a₁²+b₁²+c₁²) · √(a₂²+b₂²+c₂²))
🔹 Conditions
Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂
📏 DISTANCE OF A POINT FROM A PLANE
Vector Form:
Distance of point with position vector \vec{a} from plane \vec{r}·\vec{n} = d
Distance = |\vec{a}·\vec{n} - d|/|\vec{n}|
Cartesian Form:
Distance of point (x₁, y₁, z₁) from plane Ax + By + Cz = D
d = |Ax₁ + By₁ + Cz₁ - D|/√(A²+B²+C²)
💡 Note: Distance from origin = |d|/|\vec{n}|
📐 ANGLE BETWEEN A LINE AND A PLANE
Vector Form:
Line: \vec{r} = \vec{a} + λ\vec{b}, Plane: \vec{r}·\vec{n} = d
sin θ = |\vec{b}·\vec{n}|/(|\vec{b}| |\vec{n}|)
Cartesian Form:
Line DR's: (a, b, c), Plane: lx + my + nz + d = 0
sin θ = |al + bm + cn| / (√(a²+b²+c²) · √(l²+m²+n²))
🔹 Conditions
Line parallel to plane: al + bm + cn = 0
Line perpendicular to plane: a/l = b/m = c/n
📝 MCQ QUESTIONS
Q1. If l, m, n are direction cosines of a line, then:
(a) l² + m² + n² = 1 (b) l² + m² + n² = 0 (c) l + m + n = 1 (d) None
Answer: (a) l² + m² + n² = 1
Q2. Direction ratios of line joining (1,2,3) and (4,5,6) are:
(a) 3, 3, 3 (b) 1, 1, 1 (c) 2, 2, 2 (d) 4, 5, 6
Answer: (a) 3, 3, 3
Solution: DR = (4-1, 5-2, 6-3) = (3, 3, 3)
Q3. The equation of line passing through (1,2,3) with DR (2,3,4) is:
(a) (x-1)/2 = (y-2)/3 = (z-3)/4 (b) (x+1)/2 = (y+2)/3 = (z+3)/4 (c) (x-1)/3 = (y-2)/2 = (z-3)/4 (d) None
Answer: (a) (x-1)/2 = (y-2)/3 = (z-3)/4
Q4. Two lines with DR's (1,2,3) and (2,4,6) are:
(a) Perpendicular (b) Parallel (c) Skew (d) None
Answer: (b) Parallel
Solution: (1,2,3) and (2,4,6) are proportional (factor 2).
Q5. The angle between lines with DR's (1,0,0) and (0,1,0) is:
(a) 0° (b) 90° (c) 45° (d) 60°
Answer: (b) 90°
Solution: a₁a₂+b₁b₂+c₁c₂ = 1×0+0×1+0×0 = 0 ⇒ Perpendicular
Q6. The distance of point (1,2,3) from plane x + 2y + 2z = 10 is:
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (a) 1
Solution: d = |1+4+6-10|/√(1+4+4) = |1|/3 = 1/3
Q7. The normal vector to plane 2x + 3y + 4z = 12 is:
(a) (2,3,4) (b) (2,3,12) (c) (2,3,4) (d) (2,3,4)
Answer: (a) (2,3,4)
Q8. Planes 2x + 3y + 4z = 5 and 4x + 6y + 8z = 10 are:
(a) Perpendicular (b) Parallel (c) Same (d) None
Answer: (c) Same
Solution: Second equation is 2 × first equation.
Q9. The angle between plane 2x + 2y + z = 5 and z-axis is:
(a) 0° (b) 90° (c) 45° (d) 60°
Answer: (d) 60°
Q10. Skew lines are:
(a) Parallel (b) Intersecting (c) Non-parallel and non-intersecting (d) None
Answer: (c) Non-parallel and non-intersecting
Q11. The equation of a plane in intercept form is:
(a) x/a + y/b + z/c = 1 (b) x/a + y/b + z/c = 0 (c) ax + by + cz = 1 (d) None
Answer: (a) x/a + y/b + z/c = 1
Q12. If two lines are perpendicular, the sum of products of their DR's is:
(a) 0 (b) 1 (c) -1 (d) 2
Answer: (a) 0
Q13. The direction cosines of z-axis are:
(a) (1,0,0) (b) (0,1,0) (c) (0,0,1) (d) (1,1,1)
Answer: (c) (0,0,1)
Q14. The equation of line through origin with DR (1,2,3) is:
(a) x/1 = y/2 = z/3 (b) x/1 = y/2 = z/3 (c) x/1 = y/2 = z/3 (d) x/1 = y/2 = z/3
Answer: (a) x/1 = y/2 = z/3
Q15. The shortest distance between two skew lines is measured:
(a) Along any line (b) Along the line perpendicular to both (c) Along any parallel line (d) None
Answer: (b) Along the line perpendicular to both
✅ ANSWER KEY - MCQs
| Q.No. | Answer | Q.No. | Answer |
|---|---|---|---|
| 1 | (a) | 9 | (d) |
| 2 | (a) | 10 | (c) |
| 3 | (a) | 11 | (a) |
| 4 | (b) | 12 | (a) |
| 5 | (b) | 13 | (c) |
| 6 | (a) | 14 | (a) |
| 7 | (a) | 15 | (b) |
| 8 | (c) |
📌 KEY FORMULAS SUMMARY
| Concept | Formula |
|---|---|
| Direction Cosines | l² + m² + n² = 1 |
| Direction Ratios → DC | l = a/√(a²+b²+c²) |
| Line (Vector) | r = a + λb |
| Line (Cartesian) | (x-x₁)/a = (y-y₁)/b = (z-z₁)/c |
| Angle between Lines | cos θ = (a₁a₂+b₁b₂+c₁c₂)/√(...) |
| Shortest Distance (Skew) | |(a₂-a₁)·(b₁×b₂)|/|b₁×b₂| |
| Plane (Normal) | r·n = d |
| Plane (Intercept) | x/a + y/b + z/c = 1 |
| Distance from Plane | |Ax₁+By₁+Cz₁-D|/√(A²+B²+C²) |
| Angle between Planes | cos θ = (a₁a₂+b₁b₂+c₁c₂)/√(...) |
| Line-Plane Angle | sin θ = |al+bm+cn|/√(...) |
🔹 Quick Reference:
- Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0
- Parallel lines: a₁/a₂ = b₁/b₂ = c₁/c₂
- Perpendicular planes: a₁a₂ + b₁b₂ + c₁c₂ = 0
- Parallel planes: a₁/a₂ = b₁/b₂ = c₁/c₂
- Line parallel to plane: al + bm + cn = 0
- Line perpendicular to plane: a/l = b/m = c/n
📚 CBSE | Class XII Mathematics
Chapter 11: Three Dimensional Geometry | Study Notes