Class 12 Maths Chapter 11 Three-Dimensional Geometry

Ever thought how engineers calculate the exact slope of a flyover or how GPS systems track your location in space? All of this works because of 3D Geometry. Unlike 2D geometry, which deals only with flat surfaces, 3D geometry helps us understand points, lines, and planes in real space.

Also! The Interesting fact In 3D geometry, every point is spotted using just x, y, and z. With these, you can find distances, angles, and shortest paths between lines. This topic is super important for Class 12 boards!

Class 12 3D Geometry Formulas Notes

Too many formulas and zero clarity? Relax - these Class 12 3D Geometry notes are made exactly for quick understanding and fast revision. No heavy theory, no confusing steps - just the formulas, concepts, and logic you actually need for exams.

Whether you’re revising distances, direction cosines, planes, or skew lines, this guide keeps everything simple, structured, and easy to revise anytime - even a day before the exam.

What is 3D Geometry?

3D Geometry, as studied in Class 12 Mathematics, is the branch of geometry that deals with points, lines, and planes in three-dimensional space. Instead of two axes (x and y), we use three mutually perpendicular axes - the x-axis, y-axis, and z-axis.

A point in 3D space is represented by an ordered triplet (x, y, z). Using these coordinates, we study:

• Distance between two points
• Section formula
• Direction ratios and direction cosines
• Equations of lines and planes
• Angles between lines and planes
• Shortest distance between skew lines

All these concepts are strictly based on NCERT Class 12 syllabus.

Direction Ratios and Direction Cosines

When a line is drawn in space, it makes angles with the x-axis, y-axis, and z-axis.

• Direction cosines are the cosines of these angles.
• If the angles are α, β, γ, then direction cosines are l = cosα, m = cosβ, n = cosγ.

Important Relation

The direction cosines always satisfy: l² + m² + n² = 1

Direction Ratios

Direction ratios are numbers proportional to direction cosines. If l, m, n are direction cosines, then a, b, c can be direction ratios.

Angle Between Two Lines (Using Direction Ratios)

If two lines have direction ratios (l₁, m₁, n₁) and (l₂, m₂, n₂), then:

cosθ = (l₁l₂ + m₁m₂ + n₁n₂) ÷ √(l₁² + m₁² + n₁²) √(l₂² + m₂² + n₂²)

This formula is very important for board exams.

Equation of a Line in 3D Space

A line in 3D can be written in two common forms.

Vector Form

r = a + λb

Here:

• a is the position vector of a point on the line
• b is the direction vector

Cartesian Form

(x − x₁)/l = (y − y₁)/m = (z − z₁)/n

Where:

• (x₁, y₁, z₁) is a point on the line
• l, m, n are direction ratios

Shortest Distance Between Two Skew Lines

Skew lines are lines that:

• Do not intersect
• Are not parallel
• Do not lie in the same plane

The shortest distance between two skew lines is along the line perpendicular to both.

Formula (Vector Method)

Shortest Distance = |(a₂ − a₁) · (b₁ × b₂)| ÷ |b₁ × b₂|

This is based on the vector triple product, as given in NCERT.

Equation of a Plane

Vector Form: r · n = d

Where n is the normal vector of the plane.

Cartesian Form: ax + by + cz + d = 0

Here: (a, b, c) are direction ratios of the normal

Angle and Distance Related to Planes

• Angle between two planes depends on the angle between their normal vectors
• Angle between a line and a plane uses sine relation
• Distance of a point from a plane is calculated using a direct formula
• Distance between parallel planes depends only on constants

All these formulas are part of Class 12 3D Geometry formulas list.

Important Derivations (Exam Focus)

• Shortest distance between skew lines
• Equation of plane from vector form
• Condition for a line to intersect or be parallel to a plane

A line intersects a plane if its coordinates satisfy the plane equation. A line is parallel to a plane if its direction vector is perpendicular to the plane’s normal vector.

FAQs

Q1. What is the formula for the equation of a line in 3D geometry?

Ans. The equation of a line in 3D geometry can be written in two ways:

• Vector form: position vector of any point on the line equals position vector of a given point plus a scalar multiple of the direction vector.
• Cartesian form: the ratios of differences of x, y, and z coordinates are equal to the direction ratios of the line.

Q2. What is the equation of a plane in vector and Cartesian form?

Ans. Here is the equation of a plane in a vector and cartesian form:

• In vector form, the dot product of the position vector of a point on the plane with the normal vector is equal to a constant.
• In Cartesian form, the equation of a plane is written as ax + by + cz + d equals zero, where a, b, and c are direction ratios of the normal.

Q3. What is the formula for the angle between two lines?

Ans. The angle between two lines depends on their direction ratios. It is calculated using the dot product of their direction ratios divided by the product of their magnitudes.

Q4. What is the formula for the angle between two planes?

Ans. The angle between two planes is calculated using the normal vectors of the planes. It depends on the dot product of their normal vectors divided by the product of their magnitudes.

Q5. How do you find the distance between a point and a plane?

Ans. The distance of a point from a plane is found by substituting the point’s coordinates into the plane equation and dividing the absolute value of the result by the magnitude of the normal vector.

Q6. What is the shortest distance between two skew lines?

Ans. The shortest distance between two skew lines is the length of the perpendicular drawn between them. It is calculated using the vector triple product involving direction vectors of both lines.

Q7. What is the formula for the distance between parallel planes?

Ans. The distance between two parallel planes is the absolute difference of their constants divided by the magnitude of the normal vector.

Q8. What is the formula for the angle between a line and a plane?

Ans. The angle between a line and a plane is calculated using the direction ratios of the line and the normal vector of the plane.

Q9. How do you find the equation of a plane passing through the intersection of two planes?

Ans. The equation of a plane passing through the intersection of two given planes is obtained by taking a linear combination of the two plane equations.

Q10. What are direction cosines and direction ratios in 3D geometry?

Ans. Direction ratios are any three numbers proportional to the direction of a line, while direction cosines are the normalized values of direction ratios whose squares add up to one.