Class 12 Maths · Chapter 11
Unit: Vectors and Three-Dimensional Geometry
Tentative Study Share: 7.0 Marks
Chapter 11: Three Dimensional Geometry — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Three Dimensional Geometry.
NCERT Exercises in Chapter 11: Three Dimensional Geometry
📘 NCERT Concept Notes — Three Dimensional Geometry
01 · A direction is not a point
Direction ratios describe orientation. A position vector locates a point. Keep these roles separate when reading a line equation. Reversing all direction signs leaves the same unoriented line.
02 · One parameter, an entire line
Choose a base point and a nonzero direction. A single parameter controls all three coordinates. A zero direction component fixes that coordinate; it does not make the line invalid.
03 · Compare directions first
The angle depends on directions, not base points. Use the smaller-angle convention for unoriented lines. Orthogonal directions alone do not guarantee that spatial lines intersect.
04 · Choose the distance formula by direction
First test whether the directions are parallel. Nonparallel directions give a common normal by cross product; parallel lines need the separate height formula. The nonparallel formula also returns zero for intersection.
⚡ Key Formulas & Results — Three Dimensional Geometry
Direction cosines
How are the three direction angles encoded?
\(l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma;\quad l^2+m^2+n^2=1\)
Ratios into cosines
How do you normalise direction ratios a,b,c?
\((l,m,n)=\frac{(a,b,c)}{\sqrt{a^2+b^2+c^2}}\)
Direction from two points
Given P and Q, what direction ratios can the joining line use?
\(\vec d=\overrightarrow{PQ}=(x_2-x_1,y_2-y_1,z_2-z_1)\)
A point and a direction
What is the vector equation of a line through a with direction b?
\(\vec r=\vec a+\lambda\vec b,\quad\lambda\in\mathbb R\)
Cartesian equation
How do you eliminate the line parameter?
\(\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}\)
Smaller angle between lines
How do you ensure the angle is the smaller angle between unoriented lines?
\(\cos\theta=\frac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|},\quad0\le\theta\le\frac\pi2\)
Orthogonal directions
What equation tests whether line directions are at right angles?
\(a_1a_2+b_1b_2+c_1c_2=0\)
Parallel directions
How do you test parallel directions without dividing by a zero component?
\(\vec b_1\times\vec b_2=\vec0\quad\Longleftrightarrow\quad\vec b_2=k\vec b_1\)
Shortest distance: nonparallel lines
Which component of the displacement between the lines gives their distance?
\(d=\frac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}\)
Shortest distance: parallel lines
What replaces the common-normal formula when the directions are parallel?
\(d=\frac{|\vec b\times(\vec a_2-\vec a_1)|}{|\vec b|}\)
🎯 CBSE 2026 Board PYQs from Three Dimensional Geometry
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q16
The length of perpendicular drawn from point (2, 5, 7) on line x/1 = y/0 = z/0 is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q20
Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q35
Check whether the lines given by (x−1)/2 = (y−2)/3 = (z−3)/4 and (x−4)/5 = (y−1)/2 = z, are parallel or…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q9
Find the distance from (1,2,3) to the y-axis.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q19
Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q33
Show that the line through A=(0,−1,−1), B=(4,5,1) intersects the line through C=(3,9,4), D=(−4,4,4)…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q2
Find the perpendicular distance from (3,4,2) to the z-axis.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q20
Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q32
Show that the line through (0,4,1) and (2,3,−1) is parallel to the line through (4,5,0) and (2,6,2)…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q15
Find the direction cosines of (2x−1)/4=(1−y)/3=−z/6 in the orientation given by increasing common…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q20
Assertion: A line can have direction cosines (1,1,1). Reason: cos θ=1 is possible for θ=0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q25
Find the point on r=(0,−1,3)+λ(2,−2,1) whose coordinate sum is 3.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q35
Find the intersection of the line through A=(1,2,3), B=(5,8,11) and r=(4,1,0)+t(5,2,1). Find the line…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q8
Find the listed direction cosines of x=y=1−z.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q19
Assertion: A line can have direction cosines (1,1,1). Reason: cos θ=1 is possible for θ=0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q23
Find the foot from the origin to x=(y+1)/(−1)=(z−3)/(−2).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q33
Find the intersection of the line through A=(1,2,3), B=(5,8,11) and r=(4,1,0)+t(5,2,1). Find the line…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q1
Which listed direction cosines belong to the line x=1, y=z? The printed symmetric form is (1−x)/0=y=z.
📝 Chapterwise Practice Questions — Three Dimensional Geometry (94 Questions)
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the foot of the perpendicular from (2,−3,4) to the y-axis.
Class 12 Three Dimensional Geometry · Mcq Single Correct
A line makes angles α,β,γ with the positive coordinate axes. Find cos 2α+cos 2β+cos 2γ.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the distance of (α,β,γ) from the y-axis.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find direction cosines of (x−1)/2=(1−y)/3=(2z−1)/12.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the acute angle between two distinct body diagonals of a cube.
Class 12 Three Dimensional Geometry · Mcq Single Correct
A line makes angles θ₁,θ₂,θ₃,θ₄ with the four body diagonals of a cube. Find Σ cos²θᵢ.
Class 12 Three Dimensional Geometry · Mcq Single Correct
A directed line makes angles π/3 and π/4 with the positive x- and y-axes. What are the possible angles…
Class 12 Three Dimensional Geometry · Mcq Single Correct
P lies on the segment joining (3,2,−1) and (6,2,−2). If x(P)=5, find y(P).
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find direction cosines for angles 90°,135°,45° with the positive axes.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Which parametrization is the line through (1,1,1) parallel to the z-axis?
Class 12 Three Dimensional Geometry · Mcq Single Correct
Which equations describe the x-axis?
Class 12 Three Dimensional Geometry · Mcq Single Correct
A directed line is equally inclined to the three positive coordinate axes. Find its direction cosines.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the distance of (1,6,3) from r=(0,1,2)+t(1,2,3).
Class 12 Three Dimensional Geometry · Mcq Single Correct
Lines x=ay+b,z=cy+d and x=a′y+b′,z=c′y+d′ have orthogonal directions if and only if:
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the area of quadrilateral A(0,4,1), B(2,3,−1), C(4,5,0), D(2,6,2).
Class 12 Three Dimensional Geometry · Mcq Single Correct
If (a,b,0) lies on (x+1)/2=(y+2)/3=(z+3)/4, find (a,b).
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the shortest distance between r=(8,−9,10)+t(3,−16,7) and r=(15,29,5)+s(3,8,−5).
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the reflection of P(1,6,3) in the line x=(y−1)/2=(z−2)/2.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the angle between 2x=3y=−z and 6x=−y=−4z.
Class 12 Three Dimensional Geometry · Mcq Single Correct
Find the foot from (1,8,4) to the line joining (0,−1,3) and (2,−3,−1).
Class 12 Three Dimensional Geometry · Assertion Reason
Assertion (A): The lines through (4,7,8),(2,3,4) and through (−1,−2,1),(1,2,5) are parallel. Reason…
Class 12 Three Dimensional Geometry · Assertion Reason
Assertion (A): The line through P=(1,2,3),Q=(3,−1,3) is r=(3,−1,3)+t(2,3,0). Reason (R): The line…
Class 12 Three Dimensional Geometry · Assertion Reason
Assertion (A): For direction angles α,β,γ, sin²α+sin²β+sin²γ=2. Reason (R): Direction cosines satisfy…
Class 12 Three Dimensional Geometry · Assertion Reason
Assertion (A): The oriented unit vectors (−√3/4,1/4,−√3/2) and (−√3/4,1/4,√3/2) make an angle 120°…