Intro to Three Dimensional Geometry Class 11
Master Intro to Three Dimensional Geometry Class 11 with comprehensive NCERT Solutions, Practice Questions, MCQs, Sample Papers, Case Based Questions, and Video lessons.
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NCERT Solutions
Intro to Three Dimensional Geometry Class 11 – NCERT Solutions
Each question below opens its complete step-by-step Teachoo solution.
Ex 11.1
4 questionsEx 11.1, 1
Ex 11.1, 1 teachoo.com
A point is on the x-axis. What are its y-coordinates and z-coordinates?
If a point is on the x-axis,
then the coordinate of y and z are 0.
So, the point is (x, 0, 0).
Ex 11.1, 2
teachoo.com
Ex 11.1, 2
A point is in the XZ-plane. What can you say about its y-coordinate?
If a point is in XZ plane,
then its y-coordinate is 0.
Ex 11.1, 3
Ex 11.1, 3 teachoo.com
Name the octants in which the following points lie:
(1, 2, 3), (4, -2, 3), (4, -2, -5), (4, 2, -5), (-4, 2, -5), (4, 2, 5),
(-3, -1, 6), (2, -4, -7)
Foams) vw Lv |v [vi] viv
x + - - + + - - +
y + + - - + + - -
z + + + + - - - =
(i) (4, 2, 3)
Here x is positive, y is positive and z is positive.
So it lies in | octant.
(ii) (4, -2, 3)
Here x is positive, y is negative and z is positive.
So it lies in IV octant
Ex 11.1, 4
teachoo.com
Ex 11.1, 4
Fill in the blanks
(i) The x-axis and y-axis taken together determine a plane known
as
The x-axis and y-axis taken together determine a plane known as
XY Plane
(ii) The coordinates of points in the XY-plane are of the form .
If a point is in XY plane, then its z-coordinate is 0.
So, Coordinates are (x, y, 0)
The coordinates of points in the XY-plane are of the form (x, y, 0).
Ex 11.2
9 questionsEx 11.2, 1 (i)
Ex 11.2,1 teachoo.com
Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1)
Let P be (2, 3, 5)
and Q be (4, 3, 1)
Distance PQ = ,/ (X, — X4)* + (y. — y,)? + (@, — 24)?
Here,
x, =2,y,=3,2,=5
%=4,y,=3,2,=1
PQ = (4-—2)?+4+ (3-—3)74+(1 —5)?
= 4/22 + 07+ (-4)?
Ex 11.2, 1 (ii)
Find the distance between the following pairs of points:
(ii) (–3, 7, 2) and (2, 4, –1)
Ex 11.2, 1 (iii)
Find the distance between the following pairs of points:
(iii) (–1, 3, –4) and (1, –3, 4)
Ex 11.2, 1 (iv)
Find the distance between the following pairs of points:
(iv) (2, –1, 3) and (–2, 1, 3)
Ex 11.2,2
Ex 11.2, 2 teachoo.com
Show that points P (-2, 3, 5), Q (1, 2, 3) & R (7, 0, -1) are collinear.
If three points are collinear, then they lie ona line.
Let first calculate distance between the 3 points
i.e. PQ. QR and PR
Calculating PQ
P(-2,3,5)
Q (1, 2, 3)
Hence,
PQ=V&— x)? + G2 —y1)? + @ — 21)?
Ex 11.2, 3 (i)
Ex 11.2, 3 teachoo.com
Verify the following:
(i) (0, 7, -10), (1, 6, -6) and (4, 9, -6) are the vertices of an
isosceles triangle.
Let points be
A(0,7, —10), B(1,6, —6)&C (4,9, — 6)
If any 2 sides are equal, it will be an isosceles triangle
Lets calculate AB, BC, AC
Calculating AB
A(0, 7, — 10)
B(1, 6, —6)
AB =4/ (KX, — x1)? + 2-1)? + @ — 24)?
Ex 11.2, 3 (ii)
Verify the following:
(ii) (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right angled triangle.
Ex 11.2, 4
Ex 11.2, 4 teachoo.com
Find the equation of the set of points which are equidistant from
the points (1, 2, 3) and (3, 2,-1).
Let A (1, 2, 3) & B (3, 2,-1)
Let point P be (x, y, z,)
Let point P (x, y, z) be at equal distance from point A (1, 2, 3) &
B (3, 2,-1)
i.e. PA= PB
Calculating PA
P (x, y, 2), A (1, 2, 3)
Ex 11.2, 5
teachoo.com
Ex 11.2, 5
Find the equation of the set of points P, the sum of whose
distances from A (4, 0, 0} and B (-4, 0, 0) is equal to 10.
Given A (4,0, 0), B(-4, 0, 0)
Let the coordinates of point P be (x, y, z)
Given that,
Sum Distance of PA & PB is 10
PA + PB = 10
First, lets calculate PA & PB
Examples
14 questionsExample, 1
Example 1 teachoo.com
In Fig 12.3, if P is (2, 4, 5), find the coordinates of F.
Z
Given P (2, 4, 5)
So,x=2,y=4,z=5 c E
‘Li 7%
5)
Paint F lies in XZ plane,
4 Y
So its y-coordinate will be zero KET
A D
xX
-. Coordinates of F = (x, 0, z)
= (2, 0, 5)
Example 2
teachoo.com
Example 2
Find the octant in which the points (-3, 1, 2) and (-3, 1, - 2) lie.
Focants | 1) mh Vv Vivi
x + - - + + - - +
y + + - - + + - -
z + + + + - - - -
(i) (-3, 1,2)
Here x is negative, y is positive and z is positive
So it lies in Il octant.
(ii) (-3, 1, -2)
Here x is negative, y is positive and z is negative.
So it lies in VI octant.
Example 3
Example 3 teachoo.com
Find the distance between the points P(1, -3, 4) and Q (-4, 1, 2).
Given P (1, — 3, 4)
and Q (-4, 1, 2)
Distance PQ=,/(x.—x,)* + (v2 -—¥1)? + (2 — 2)?
Here,
X,=1,y, = -3,2,=4
%=-4,y =1, 2,=2
pa=f/(—4— 17+ 1 — (-3))? + 2-4)?
=/(-5) + (1+ 3)? + (-2)
Example 4
Example 4 teachoo.com
Show that the points P (—2, 3, 5), Q (1, 2, 3) and R (7, 0, -1) are
collinear.
If three points are collinear, then they lie ona line.
Let first calculate distance between the 3 points
i.e. PQ. QR and PR
Calculating PQ
P(-2,3,5)
Q (1, 2, 3)
Hence,
PQ=V¥ G2 - x1 + On - V1)? + — 24)?
Example 5
Example 5 teachoo.com
Are the points A (3, 6, 9), B (10, 20, 30) and C (25, — 41, 5), the
vertices of a right angled triangle?
Lets first calculate distances AB, BC and AC
& then apply Pythagoras theorem to check whether it is right triangle}
Calculating AB
A (3, 6, 9)
B (10, 20, 30)
AB= VG — x1)? + G2 — 1)? + @, — 24)"
Here, x, =3, y,=6,2,=9
x, = 10, y, = 20, z, = 30
Example 6
Example 6 teachoo.com
Find the equation of set of points P such that PA? + PB? = 2k?, where
A and B are the points (3, 4, 5) and (—1, 3, -7), respectively.
Given A (3, 4, 5)
&B(-1,3, -7)
Let the co-ordinates of point P be (x, y, z)
We need to find equation of set of point P (x, y, z)
Such that PA? + PB2=2k? _...{1)
First, we calculate (PA), (PB)*
Example, 7
teachoo.
Example 7 ‘eachoo.com
Show that the points A (1, 2, 3), B (-1, -2, -1), C (2, 3, 2) & D (4, 7, 6)
are the vertices of a parallelogram ABCD, but it is not a rectangle.
Difference between
Rectangle Parallelogram
Opposite sides are Opposite sides are
equal equal
+
Diagonals are equal 0 (4,7, 6) C2, 3, 2)
We need to prove that ABCD is a ></
llel but not tangl A 8
parallelogram but not a rectangle (1, 2, 3) (1, -2,-1)
Example 8
Example 8 teachoo.com
Find the equation of the set of the points P such that its distances
from the points A (3, 4, -5) and B (— 2, 1, 4) are equal.
Given A (3, 4, -5) & B(-2, 1, 4)
Let point P be (x, y, z,)
Given PA= PB
Calculating PA
PA = V/ (K — x)? + G2—-y1)?+ @ — 21)
Here, X,=X,y, =Y, 2, =Z
%=3,y.=4, 2,=-5
Example 9
Example 9 teachoo.com
The centroid of a triangle ABC is at the point (1, 1, 1). If the
coordinates of A and B are (3, -5, 7) and (-1, 7, -6}, respectively,
find the coordinates of the point C.
A (3,-5, 7)
Let ABC be a triangle .
where A (3, -5, 7}, B(-1, 7, -6}
Let G be the centroid of A ABC
So, G (1, 1, 1) B(-1, 7, -6) Che ¥, 2)
Let Coordinate of C (x, y, z}
Question 1 (i)
teachoo.com
Example 7
Find the coordinates of the point which divides the line segment
joining the points (1, —2, 3) and (3, 4, -5) in the ratio 2:3
(i) internally,
Let the 2 given points be e 2 - >
A P B
A(t 2, 3) (1-23) v2) (3,4,-5)
& B (3, 4,5}
Let P (x, y, z,) be points that divides line in ratio 2:3 internally
Question 1 (ii)
Example 7
Find the coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3
(ii) externally,
Question 2
Example 8 teachoo.com
Using section formula, prove that the three points (— 4, 6, 10),
(2, 4, 6) and (14, 0, —2) are collinear.
Let points be
A(—4,6, 10), B (2, 4, 6), C (14, 0,—2)
Point A, B, & C are collinear if point C divides AB in some ratio
externally & internally
We know that
Co-ordinate of point P(x, y ,z) that divides line segment joining
(Xp) Var 21) & (Xp, Yo Z2) in ration m:n is
Question 3
Example 9 teackoo.com
Find the coordinates of the centroid of the triangle whose vertices
are (x1, ¥z, 21), (Xp, Yor Zo) and (x3, Ys, Zs).
Let ABC be the triangle where A(X. Vu 24)
A (Xt, Var 21), BOX, Yo, 22) and C(x, Y3, Zs)
We need to find co-ordinate of centroid.
Let G be the centroid of A ABC
B(Xp, Yor Za) D C(X3, Var Z3)
Let AD be the median of A ABC
So, Dis the mid point of BC
Question 4
teachoo.com
Example 10
Find the ratio in which the line segment joining the points (4, 8, 10}
and (6, 10, —8) is divided by the YZ-plane.
Let AB be the line segment
joining points A (4, 8, 10) & (6, 10, —8}
k 1
o_e___-®
Let YZ Plane divide line AB A P B
at P (x, y, z) in the ratio k: 1 (4,8, 10) (0, y,2) (6, 10, -8)
Co-ordinate of P that divide line segment joining point A (x, y,, Z,) &
B((X, Yo, Z)) in the ratio m : nis
_ (“= tnx, my,+ny, mz,+ 21)
“Xi mtn ’ mtn ’ mtn
Miscellaneous
6 questionsMisc 1
Misc 1 teachoo.com
Three vertices of a parallelogram ABCD are A (3, -1, 2), B (1, 2, -4)
and C (-1, 1, 2). Find the coordinates of the fourth vertex.
Let D be (x, y, z}
In parallelogram, diagonals bisect each other
» AO=OC D (x,y, 2) C(-1, 1, 2)
.
We can say that Vax
A{3, -1, 2 B (1, 2, -4
Ois Midpoint of AC (3,-1, 2) (1,2, ~4)
and
O is Midpoint of BD
Misc 2
Misc 2 teachoo.com
Find the lengths of the medians of the triangle with vertices A (0, 0, 6),
B (0, 4, 0) and (6, 0, 0).
Let A ABC where A(O, 0, 6)
AD, BE, CF are medians
F E
Since median bisects the opposite side
Dis Midpoint of BC
Eis Midpoint of AC D
B(0, 4, 0) C(6, 0, 0}
F Midpoint of AB
Dis midpoint of BC Eis midpoint of AC Fis midpoint of AB
0+6 440 04+0 0+6 040 6+0 0+0 044 640
D= (APS) | OF GEES) oe SS)
D = (3, 2, 0) D = (3, 0, 3) D = (0, 2, 3)
Misc 3
Misc 3 teachoo.com
If origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q
(-4, 3b, -10} and R (8, 14, 2c}, then find the values of a, b and c.
Given A PQR where P (2a, 2, 6)
P (2a, 2, 6), Q (-4, 3b, -10} , R (8, 14, 2c}
Also, e
Origin O (0, 0, 0) is the centroid of APQR 0 (0, 0, 0)
Q (-4, 3b, -10) R (8, 14, 2c
We know that
Co ordinate of centroid whose vertices are
(Xa, Var 24), (Xa, Var Za), (Xz, Yo, Z3) is
@ +¥1 +41 X2t¥ot42 X%3 + Y3 *2)
3 , 3 , 3
Misc 4
Misc 4 teachoo.com
If A and B be the points (3, 4, 5) and (-1, 3, -7), respectively, find
the equation of the set of points P such that PA? + PB? = k*, where k
is a constant.
Given points
A (3, 4, 5) & B(—1, 3,-7)
Let Point P be (x, y, z,)
We need to point equation of points P,
such that PA? + PB*=k? ...(1)
Question 1
Misc 4 teachoo.com
Find the coordinates of a point on y-axis which are at a distance of
5¥2 from the point P (3, —2, 5).
Since point is on y-axis
its x & z coordinate is 0
Let the point on y-axis be A(O, a, 0)
Given that
Point A is at a distance of 5V2 from point P (3, — 2, 5)
ie. PA= 52
Finding Distance PA
Question 2
Misc 5 teachoo.com
A point R with x-coordinate 4 lies on the line segment joining the
points P (2, —3, 4) and Q (8, 0, 10). Find the coordinates of the point R.
[Hint suppose R divides PQ in the ratio k: 1. The coordinates of the
int R . b (Ee -3 e+ 4),
point R are given by (7
Given that
Point R lie on the line segment PQ.
and x — Coordinate of Ris 4
k 1
o—___e__-®
P R Q
Let Point R be (4, b, c) (2,-3,4) (4 y,z) (8,0, 10)
Let R divide line segment PR in the ratio k: 1
Section Formula in 3D Geometry
6 questionsQuestion 1 (i)
Ex 12.3,1 teachoo.com
Find the coordinates of the point which divides the line segment
joining the points (—2, 3, 5) and (1, —4, 6) in the ratio
(i) 2:3 internally.
Let A be (-2, 3, 5) 2 3
o——__e—___e
&Bbe (1,4, 6) A P B
(-2,3,5) (x y,z) (1, -4, 6)
Let coordinate of point P be (x, y, z) that divides the line joining A
& B in the ratio of 2 : 3 internally
We know that
Coordinate of P that divide the line segment joining
A(X, Vp Z1) & B(X>, Yo, Z2) internally in the ratio m: nis
MXgtNxX, Myzt+nNyYy Mzg+nZy,
P(x, y 2) = | 7 >»,
mtn mtn mtn
Question 1 (ii)
Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (ii) 2:3 externally.
View solutionQuestion 2
teachoo.com
Ex 12.3, 2
Given that P (3, 2, —4), Q (5, 4, -6) and R (9, 8, -10) are collinear. Find
the ratio in which Q divides PR.
Given that
Point P (3, 2, —4), Q (5, 4, -6) & R (9, 8, -10) are collinear
Q must divide line segment PR in some ratio externally & internally .
We know that
Co-ordinate of point P(x, y ,z) that divides line segment joining
(Xi, Var 21) & (Xq, Yo Z2) in ration m:n is
(x,y,z) = (oe +nx, ; my,+ ny, ; mMzZ2 + ™1)
mtn mtn mtn
Question 3
Ex 12.3, 3 teachoo.com
Find the ratio in which the YZ-plane divides the line segment
formed by joining the points (—2, 4, 7) and (3, —5, 8).
Let AB be the line segment
joining points A {—2, 4, 7) & B (3, -5, 8)
k 1
Let YZ Plane divide line AB A Pp B
at P (x, y, z) in the ratio k: 1 (-2,4,7) (0, y,z) (3, -5, 8)
Co-ordinate of P that divide line segment joining point A (x, y,, Z,)
& B((x,, Yo Z.) in the ratio m : nis
_ (= nx, my,tny, mz,+ ~21)
“\imtn ’ mtn’ mtn
Question 4
Ex 12.3, 4 teachoo.com
Using section formula, show that the points A (2, —3, 4), B (-1, 2, 1)
and C (0,5, 2) are collinear.
3
Given Points A (2, -3, 4), B(-1, 2, 1) &C (0, * 2)
Point A, B & C are collinear if ,
point C divides AB in some ratio externally or internally.
We know that
Co-ordinate of point P(x, y ,z) that divides line segment joining (x,
Vp 21) & (0%, Yo, Z.) in ration m:n is
(x, y,2) = (= +nx, ; my, + ny, ; mz2 + a)
mtn mtn mtn
Question 5
Ex 12.3, 5 teachoo.com
Find the coordinates of the points which trisect the line segment
joining the points P (4, 2, -6) and Q (10, —16, 6).
Let Point A (a, b, c} & point B (p, q, r) trisect the
line segment PQ.
1 1 1
o——_e—__—__e—_e
P A B Q
(4,2,-6) (a,b,c) — (p,q,r) (10, -16, 6)
i.e. PA=AB=BC
Point A divides PQ in the ratio of 1: 2
1 2
o—_e—__—_________
P A Q
(4,2,-6) (a,b,c) (10, -16, 6)
Why Learn This With Teachoo?
Introduction to Three-Dimensional Geometry extends coordinate methods from a plane to space. Students learn the three coordinate axes, coordinate planes, octants, signs of coordinates, distance between two points and section formulas. Teachoo provides solutions for Exercises 11.1 and 11.2, NCERT examples, miscellaneous questions and concept-wise practice involving definition, distance, collinearity, verification, loci, internal division, centroid and median.
Coordinates and octants in space
Three mutually perpendicular axes—x, y and z—meet at the origin O. A point P is represented by an ordered triple (x, y, z). The xy-plane has z = 0, the yz-plane has x = 0 and the zx-plane has y = 0. A point on an axis has two coordinates equal to zero.
The coordinate planes divide space into eight octants. Signs of x, y and z identify the octant. Students should visualise the projection of a point onto the coordinate planes and distinguish an ordered triple from an ordered pair. Changing the order changes the point.
Distance between two points
For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂),
PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
This is the three-dimensional extension of the Pythagorean distance formula. Distance from the origin to (x, y, z) is √(x² + y² + z²). Squared distances are often sufficient when testing equality or collinearity and avoid unnecessary radicals.
Distance conditions can describe a set of points. For example, points equidistant from two fixed points satisfy an equation that simplifies to a plane. Students also verify geometric properties by comparing side lengths.
Section formula and centroid
If P divides the segment joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) internally in the ratio m, where AP = m, then
P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n)).
For external division, the corresponding denominator becomes m − n with appropriate directed weighting. The midpoint is obtained by m = n = 1. The centroid of a triangle with vertices A, B and C is the coordinate-wise average: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3).
Students use section ideas to prove collinearity, locate a median point and verify that the centroid divides each median in the ratio 2:1.
Topics covered on Teachoo
-
Exercises 11.1 and 11.2, examples and miscellaneous questions;
-
definition of 3D coordinates;
-
axes, coordinate planes and octants;
-
distance formula and its derivation;
-
distance-based collinearity and verification;
-
sets of points satisfying distance conditions;
-
section formula and internal division;
-
plane dividing a line segment;
-
centroid and median questions.
Learning outcomes
Students should be able to locate points in space, identify coordinate planes and determine octant signs. They should calculate distances, verify geometric relationships and formulate simple distance loci. They should apply the section formula, find midpoints and centroids and use coordinates to test collinearity.
Why is this chapter important?
Three-dimensional coordinates are essential for Class 12 vectors and 3D geometry. They also appear in physics, engineering, computer graphics and spatial data. This introductory chapter builds spatial reasoning while using familiar algebraic tools.
The distance and section formulas also demonstrate an important mathematical principle: many plane results generalise by adding an independent coordinate. Recognising this structure makes later vector formulas less intimidating. Instead of treating 3D as an entirely new subject, students can view it as coordinate geometry with one additional perpendicular direction.
How to approach proof and parameter questions
When an unknown coordinate or parameter appears, translate the geometric condition into an equation. Equal distances can be compared after squaring both sides, collinearity can be tested through section ratios, and a midpoint condition becomes three coordinate equations. Solve the resulting algebra and then verify every possible value in the original geometric condition, especially if squaring was used.
How Teachoo helps you prepare
Teachoo separates distance and section questions into definition, collinearity, verification and application groups. Begin with labelled axes and practise the signs in each octant. When using a formula, align x-coordinates with x-coordinates, and repeat independently for y and z.
For ratio questions, write AP = m next to the formula to prevent reversed weights. Use serial-order NCERT solutions for textbook completion and concept-wise practice for centroids, medians and set-of-points questions.
School-exam, JEE and competency preparation
School exams test coordinates, distances, ratios and centroids. Competitive questions may combine distance equations, unknown parameters and geometric conditions. Compare squared distances when possible and solve the resulting algebra only after expanding carefully.
Competency questions may model locations, rooms, navigation or 3D objects. Fix the origin and axes explicitly. Interpret zeros correctly: a zero coordinate places a point on a plane, not necessarily at the origin. If a point lies between two endpoints, its coordinates should usually lie between corresponding endpoint coordinates; use this as a reasonableness check.
Quick revision checklist
Identify axes and planes from coordinate conditions; list sign patterns for eight octants; calculate three distances; verify an isosceles or right triangle; find two internal division points; calculate a midpoint and centroid; and solve one distance-locus problem.
Common mistakes to avoid
Do not omit the z-coordinate in a distance calculation. Do not reverse the ratio weights in the section formula. A point with z = 0 lies in the xy-plane, not on the z-axis. Collinearity requires more than one equal distance; show that one distance equals the sum of the other two or use consistent ratios.
Deeper reasoning and concept connections
Study Introduction to Three-Dimensional Geometry through comparison and justification. Place two related examples side by side, identify the decisive difference and explain why one method works in each case. Then create a new example and a deliberate non-example. This forces the definition to do real work and exposes gaps that passive reading hides.
Students should also practise reversing questions. After solving for an answer, ask what question could have produced it, whether more than one answer is possible and which extra condition would make the result unique. Reverse reasoning develops flexibility and is especially useful for missing-value, assertion–reason and error-analysis questions. The goal is to understand the network of ideas, not merely the order of a textbook solution.
How to solve unfamiliar and competency-based questions
Begin by separating facts from conclusions. Facts are given by the question or a known property; conclusions must be derived. Draw or rewrite the problem so each fact has a visible place. If several methods are possible, prefer the one with fewer assumptions and an easy final check. Record intermediate results rather than doing everything mentally.
Competency questions often change context without changing mathematics. Replace names and story details with variables, shapes, sets or data values. After solving, restore the context and check feasibility: counts should be whole where required, lengths and areas should have suitable units, probabilities should lie between 0 and 1, and constructed figures should satisfy every stated condition.
What complete mastery looks like
For Introduction to Three-Dimensional Geometry, a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.
Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.
Additional frequently asked questions
What should a student know before starting Introduction to Three-Dimensional Geometry?
Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.
How can a student check an answer in Introduction to Three-Dimensional Geometry?
Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.
How many questions are enough for strong preparation?
There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.
How should Teachoo solutions be used without becoming dependent on them?
Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.
Frequently asked questions
How many octants are there in three-dimensional coordinate geometry?
The three coordinate planes divide space into eight octants.
What is the distance from the origin to (x, y, z)?
It is √(x² + y² + z²).
How is the 3D section formula different from the 2D formula?
The same weighted-average idea is applied to three coordinates instead of two.
What are the coordinates of a triangle’s centroid in space?
They are the coordinate-wise averages of the three vertices.
What does Teachoo cover beyond basic formulas?
Teachoo includes collinearity, verification, distance-locus, plane-division, centroid and median questions alongside NCERT solutions.
Visualise the point first and calculate second. Keeping the three coordinate directions separate makes 3D geometry a natural extension of coordinate geometry in a plane.