Three Dimensional Geometry

part 1

Direction Cosines and Lines

• Direction cosines and direction ratios
• Equation of a line — vector and Cartesian form
• Line passing through two points
• Angle between two lines

part 2

Distance and Skew Lines

• Shortest distance between two lines
• Distance between skew lines
• Distance between parallel lines
• Coplanarity of lines

part 3

Planes

• Equation of a plane — vector and Cartesian form
• Angle between two planes
• Angle between line and plane
• Distance of point from plane
• Equation of plane through intersection of two planes

Vector equation of line

$\vec{r} = \vec{a} + \lambda\vec{b}$

★ board favourite

Cartesian equation of line

$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$

★ board favourite

Angle between two lines

$\cos\theta = \frac{|a_1a_2+b_1b_2+c_1c_2|}{|\vec{b}_1||\vec{b}_2|}$

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Lines perpendicular

$a_1a_2 + b_1b_2 + c_1c_2 = 0$

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Shortest distance (skew lines)

$d = \frac{|(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}$

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Vector equation of plane

$\vec{r}\cdot\hat{n} = d$

★ board favourite

Cartesian equation of plane

$ax + by + cz = d$

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Plane through point with normal (a,b,c)

$a(x-x_1)+b(y-y_1)+c(z-z_1) = 0$

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Distance from point to plane

$\text{dist} = \frac{|ax_1+by_1+cz_1-d|}{\sqrt{a^2+b^2+c^2}}$

★ board favourite

Angle between two planes

$\cos\theta = \frac{|\vec{n}_1\cdot\vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$

★ board favourite

Find the coordinates of the foot of the perpendicular and the length of the perpendicular drawn from the point (1, 2, 3) to the plane 2x − y + z + 1 = 0. Also find the image of the point in the plane.

Find the angle between the pair of lines (x−2)/2 = (y−1)/5 = (z+3)/−3 and (x+2)/−1 = (y−4)/8 = (z−5)/4.

Find the equation of the plane passing through (1, 2, 3), (2, 3, 1), and (3, 1, 2). Also find the distance of the point (0, 0, 0) from this plane.

Show that the lines (x+3)/−3 = (y−1)/1 = (z−5)/5 and (x+1)/−1 = (y−2)/2 = (z−5)/5 are coplanar. Also find the equation of the plane containing them.

Find the shortest distance between the lines: r⃗ = (î + ĵ) + λ(2î − ĵ + k̂) and r⃗ = (2î + ĵ − k̂) + μ(3î − 5ĵ + 2k̂).

Find the equation of the line passing through the point (−1, 3, −2) and perpendicular to the two lines x/1 = y/2 = z/3 and (x+2)/−3 = (y−1)/2 = (z+1)/5.

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

Find the angle between the line (x+1)/2 = y/3 = (z−3)/6 and the plane 10x + 2y − 11z = 3.

Find the vector equation of the plane which is at a distance of 6/√29 units from the origin and has 2î − 3ĵ + 4k̂ as the normal vector.

Find the distance of the point (−1, −5, −10) from the point of intersection of the line r⃗ = 2î − ĵ + 2k̂ + λ(3î + 4ĵ + 2k̂) and the plane r⃗ · (î − ĵ + k̂) = 5.

How many marks is Three Dimensional Geometry worth in the Class 12 board exam?

Around 8 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Equation of plane, angle between line and plane — multi-step.

Is Three Dimensional Geometry easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 4 hours to cover it well.

What are the important topics in Three Dimensional Geometry?

The chapter covers 13 NCERT topics in 3 areas: Direction Cosines and Lines; Distance and Skew Lines; Planes.

What questions come from Three Dimensional Geometry in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Line and Plane, Angle between Lines, Planes, Coplanar Lines — 1- to 5-mark questions.