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Mathematics · Class 11 Science

Ch 5Introduction to Three-Dimensional Geometry — Class 11 Mathematics, concept-first.

You already know how to fix a point's position on a flat surface like a sheet of paper. You draw two perpendicular lines — the x‑axis and the y‑axis — and then describe the point by two numbers: its perpendicular distances from these axes. Those two numbers are its coordinates.

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Coordinate Axes Equations

Imagine a blank sheet of paper. You want to describe exactly where a point is. The most natural way is to draw two perpendicular lines — one horizontal, one vertical — and measure distances from them.

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8 concepts · 135 appearances

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

11.1

Introduction

You already know how to fix a point's position on a flat surface like a sheet of paper. You draw two perpendicular lines — the x‑axis and the y‑axis — and then describe the point by two numbers: its p…

11.2

Coordinate Axes and Coordinate Planes in Three Dimensional Space

We begin with a point in space. Through , imagine three planes that are all mutually perpendicular — each plane meets the other two at right angles.

11.3

Coordinates of a Point in Space

6 Q

Once we have fixed a coordinate system — the three mutually perpendicular axes (x, y, z), the three coordinate planes (XY, YZ, ZX), and the origin O — the next task is to connect every point in space…

+Worked Examplesi2 questions
  1. Example 1A rectangular parallelepiped (box) is drawn with its edges parallel to the coordinate axes, one vertex at the origin $O(0,0,0)$ and the diag…Free
  2. Example 2Find the octant in which the points $(-3, 1, 2)$ and $(-3, 1, -2)$ lie.Preview
+Exercise 11.1i4 questions
  1. Q1A point is on the $x$-axis. What are its $y$-coordinate and $z$-coordinates?Free
  2. Q2A point is in the XZ-plane. What can you say about its $y$-coordinate?Free
  3. Q3Name the octants in which the following points lie: $(1, 2, 3)$, $(4, -2, 3)$, $(4, -2, -5)$, $(4, 2, -5)$, $(-4, 2, -5)$, $(-4, 2, 5)$, $(-…Preview
  4. Q4Fill in the blanks: (i) The $x$-axis and $y$-axis taken together determine a plane known as _______. (ii) The coordinates of points in the X…Preview
11.4

Distance Between Two Points

9 Q

We already know how to find the distance between two points on a plane, using the familiar formula . The step into three dimensions is a natural extension of that idea.

+Worked Examplesi4 questions
  1. Example 3Find the distance between the points $P(1, -3, 4)$ and $Q(-4, 1, 2)$.Free
  2. Example 4Show that the points $P(-2, 3, 5)$, $Q(1, 2, 3)$ and $R(7, 0, -1)$ are collinear.Free
  3. Example 5Are the points $A(3, 6, 9)$, $B(10, 20, 30)$ and $C(25, -41, 5)$, the vertices of a right angled triangle?Preview
  4. Example 6Find the equation of set of points $P$ such that $PA^2 + PB^2 = 2k^2$, where $A$ and $B$ are the points $(3, 4, 5)$ and $(-1, 3, -7)$, respe…Preview
+Exercise 11.2i5 questions
  1. Q1Find the distance between the following pairs of points: (i) $(2, 3, 5)$ and $(4, 3, 1)$ (ii) $(-3, 7, 2)$ and $(2, 4, -1)$ (iii) $(-1, 3, -…Free
  2. Q2Show that the points $(-2, 3, 5)$, $(1, 2, 3)$ and $(7, 0, -1)$ are collinear.Free
  3. Q3Verify the following: (i) $(0, 7, -10)$, $(1, 6, -6)$ and $(4, 9, -6)$ are the vertices of an isosceles triangle. (ii) $(0, 7, 10)$, $(-1, 6…Preview
  4. Q4Find the equation of the set of points which are equidistant from the points $(1, 2, 3)$ and $(3, 2, -1)$.Preview
  5. Q5Find 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 $10$.Preview

Miscellaneous Examples

+Miscellaneous Examplesi3 questions
  1. Example 7Show that the points $A(1, 2, 3)$, $B(-1, -2, -1)$, $C(2, 3, 2)$ and $D(4, 7, 6)$ are the vertices of a parallelogram $ABCD$, but it is not…Free
  2. Example 8Find 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.Preview
  3. Example 9The centroid of a triangle $ABC$ is at the point $(1, 1, 1)$. If the coordinates of $A$ and $B$ are $(3, -5, 7)$ and $(-1, 7, -6)$, respecti…Preview

Miscellaneous Exercise on Chapter 11

+Miscellaneous Exercisei4 questions
  1. Q1Three 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.Free
  2. Q2Find the lengths of the medians of the triangle with vertices $A(0, 0, 6)$, $B(0, 4, 0)$ and $(6, 0, 0)$.Free
  3. Q3If the 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 o…Preview
  4. Q4If $A$ 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^2 + PB^2…Preview

Summary

- The coordinate axes (, , ) are mutually perpendicular lines through the origin . The coordinate planes are -plane (), -plane (), and -plane (). Any point in space is written as an ordered triple .

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 50 questions50 questions
  1. Q1Locate the following points: (i) $(1,-1,3)$, (ii) $(-1,2,4)$, (iii) $(-2,-4,-7)$, (iv) $(-4,2,-5)$.Free
  2. Q2Name the octant in which each of the following points lies: (i) $(1,2,3)$, (ii) $(4,-2,3)$, (iii) $(4,-2,-5)$, (iv) $(4,2,-5)$, (v) $(-4,2,5…Free
  3. Q3Let A, B, C be the feet of perpendiculars from a point P on the $x$, $y$, $z$-axis respectively. Find the coordinates of A, B and C in each…Free
  4. Q4Let A, B, C be the feet of perpendiculars from a point P on the $xy$, $yz$ and $zx$-planes respectively. Find the coordinates of A, B, C in…Preview
  5. Q5How far apart are the points $(2,0,0)$ and $(-3,0,0)$?Preview
  6. Q6Find the distance from the origin to $(6,6,7)$.Preview
  7. Q7Show that if $x^2+y^2=1$, then the point $\left(x,\ y,\ \sqrt{1-x^2-y^2}\right)$ is at a distance 1 unit from the origin.Preview
  8. Q8Show that the point $A(1,-1,3)$, $B(2,-4,5)$ and $(5,-13,11)$ are collinear.Preview
  9. Q9Three consecutive vertices of a parallelogram ABCD are $A(6,-2,4)$, $B(2,4,-8)$, $C(-2,2,4)$. Find the coordinates of the fourth vertex. [Hi…Preview
  10. Q10Show that the triangle ABC with vertices $A(0,4,1)$, $B(2,3,-1)$ and $C(4,5,0)$ is right angled.Preview
  11. Q11Find the third vertex of triangle whose centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,-5)$.Preview
  12. Q12Find the centroid of a triangle, the mid-point of whose sides are $D(1,2,-3)$, $E(3,0,1)$ and $F(-1,1,-4)$.Preview
  13. Q13The mid-points of the sides of a triangle are $(5,7,11)$, $(0,8,5)$ and $(2,3,-1)$. Find its vertices.Preview
  14. Q14Three vertices of a Parallelogram ABCD are $A(1,2,3)$, $B(-1,-2,-1)$ and $C(2,3,2)$. Find the fourth vertex D.Preview
  15. Q15Find the coordinate of the points which trisect the line segment joining the points $A(2,1,-3)$ and $B(5,-8,3)$.Preview
  16. Q16If the origin is the centriod of a triangle ABC having vertices $A(a,1,3)$, $B(-2,b,-5)$ and $C(4,7,c)$, find the values of $a$, $b$, $c$.Preview
  17. Q17Let $A(2,2,-3)$, $B(5,6,9)$ and $C(2,7,9)$ be the vertices of a triangle. The internal bisector of the angle A meets BC at the point D. Find…Preview
  18. Q18Show that the three points $A(2,3,4)$, $B(-1,2,-3)$ and $C(-4,1,-10)$ are collinear and find the ratio in which C divides AB.Preview
  19. Q19The mid-point of the sides of a triangle are $(1,5,-1)$, $(0,4,-2)$ and $(2,3,4)$. Find its vertices. Also find the centriod of the triangle…Preview
  20. Q20Prove that the points $(0,-1,-7)$, $(2,1,-9)$ and $(6,5,-13)$ are collinear. Find the ratio in which the first point divides the join of the…Preview
  21. Q21What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edge…Preview
  22. Q22The distance of point $P(3,4,5)$ from the $yz$-plane is (A) 3 units (B) 4 units (C) 5 units (D) 550Preview
  23. Q23What is the length of foot of perpendicular drawn from the point $P(3,4,5)$ on $y$-axis (A) $\sqrt{41}$ (B) $\sqrt{34}$ (C) $5$ (D) none of…Preview
  24. Q24Distance of the point $(3,4,5)$ from the origin $(0,0,0)$ is (A) $\sqrt{50}$ (B) $3$ (C) $4$ (D) $5$Preview
  25. Q25If the distance between the points $(a,0,1)$ and $(0,1,2)$ is $\sqrt{27}$, then the value of $a$ is (A) $5$ (B) $\pm 5$ (C) $-5$ (D) none of…Preview
  26. Q26$x$-axis is the intersection of two planes (A) $xy$ and $xz$ (B) $yz$ and $zx$ (C) $xy$ and $yz$ (D) none of thesePreview
  27. Q27Equation of $y$-axis is considered as (A) $x=0,\ y=0$ (B) $y=0,\ z=0$ (C) $z=0,\ x=0$ (D) none of thesePreview
  28. Q28The point $(-2,-3,-4)$ lies in the (A) First octant (B) Seventh octant (C) Second octant (D) Eighth octantPreview
  29. Q29A plane is parallel to $yz$-plane so it is perpendicular to : (A) $x$-axis (B) $y$-axis (C) $z$-axis (D) none of thesePreview
  30. Q30The locus of a point for which $y=0,\ z=0$ is (A) equation of $x$-axis (B) equation of $y$-axis (C) equation at $z$-axis (D) none of thesePreview
  31. Q31The locus of a point for which $x=0$ is (A) $xy$-plane (B) $yz$-plane (C) $zx$-plane (D) none of thesePreview
  32. Q32If a parallelopiped is formed by planes drawn through the points $(5,8,10)$ and $(3,6,8)$ parallel to the coordinate planes, then the length…Preview
  33. Q33L is the foot of the perpendicular drawn from a point $P(3,4,5)$ on the $xy$-plane. The coordinates of point L are (A) $(3,0,0)$ (B) $(0,4,5…Preview
  34. Q34L is the foot of the perpendicular drawn from a point $(3,4,5)$ on $x$-axis. The coordinates of L are (A) $(3,0,0)$ (B) $(0,4,0)$ (C) $(0,0,…Preview
  35. Q35The three axes OX, OY, OZ determine ________ .Preview
  36. Q36The three planes determine a rectangular parallelopiped which has ________ of rectangular faces.Preview
  37. Q37The coordinates of a point are the perpendicular distance from the ________ on the respectives axes.Preview
  38. Q38The three coordinate planes divide the space into ________ parts.Preview
  39. Q39If a point P lies in $yz$-plane, then the coordinates of a point on $yz$-plane is of the form ________.Preview
  40. Q40The equation of $yz$-plane is ________.Preview
  41. Q41If the point P lies on $z$-axis, then coordinates of P are of the form ________.Preview
  42. Q42The equation of $z$-axis, are ________.Preview
  43. Q43A line is parallel to $xy$-plane if all the points on the line have equal ________.Preview
  44. Q44A line is parallel to $x$-axis if all the points on the line have equal ________.Preview
  45. Q45$x=a$ represent a plane parallel to ________.Preview
  46. Q46The plane parallel to $yz$-plane is perpendicular to ________.Preview
  47. Q47The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are…Preview
  48. Q48If the distance between the points $(a,2,1)$ and $(1,-1,1)$ is 5, then $a$ _______.Preview
  49. Q49If the mid-points of the sides of a triangle AB; BC; CA are $D(1,2,-3)$, $E(3,0,1)$ and $F(-1,1,-4)$, then the centriod of the triangle ABC…Preview
  50. Q50Match each item given under the column $C_1$ to its correct answer given under column $C_2$. Column $C_1$: (a) In $xy$-plane; (b) Point $(2,…Preview