NCERT Solutions for Class 11 Maths Chapter 11 Exercise 11.1 – Introduction to Three Dimensional Geometry introduces students to the basic concepts of 3D coordinate geometry. This exercise covers the coordinate axes, coordinate planes, and the eight octants formed by the three coordinate planes in space.
Prepared according to the latest CBSE Class 11 Maths syllabus 2025-26, Exercise 11.1 helps students understand how to locate a point in three-dimensional space using its x, y, and z coordinates. Students also learn to identify which octant a given point lies in based on the signs of its coordinates.
NCERT Solutions for Class 11 Maths – Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.1
Q.
A point is on the x -axis. What are its y-coordinate and z-coordinates?
Q.
A point is in the XZ-plane. What can you say about its y-coordinate?
Q.
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).
Q.
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as_______.
(ii) The coordinates of points in the XY-plane are of the form _______.
(iii) Coordinate planes divide the space into ______ octants.
Q.
Verify 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, 6) and (– 4, 9, 6) are the vertices of a right angled triangle.
(iii) (–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.
Q.
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Q.
Find the coordinates of the points which trisect the line segment joining the points P(4, 2, – 6) and Q(10, –16, 6).
Q.
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.
Q.
Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).
Q.
If 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 of a, b and c.
Q.
Find the coordinates of a point ony−axis whichare at a distance of 52fromthe point P (3,−2,5).
Q.
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.
Q.
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 PA2 + PB2 = k2, where k is a constant.
NCERT Solutions for Class 11 Maths – Chapter 11 Introduction to Three Dimensional Geometry Exercise 11.1
The solutions are explained in a clear, step-by-step format to help students build a strong foundation in 3D geometry, which is essential for higher studies in mathematics and physics.
Q1. A point is on the x-axis. What are its y-coordinate and z-coordinates?
Ans: The coordinates of any point on the x-axis are of the form (x, 0, 0).
Therefore, if a point is on the x-axis, its y-coordinate = 0 and z-coordinate = 0.
Q2. A point is in the XZ-plane. What can you say about its y-coordinate?
Ans: The coordinates of any point in the XZ-plane are of the form (x, 0, z).
Therefore, if a point is in the XZ-plane, its y-coordinate is always 0.
Q3. Name the octants in which the following points lie:
The sign convention for octants is:
| Coordinate |
I |
II |
III |
IV |
V |
VI |
VII |
VIII |
| x |
+ |
− |
− |
+ |
+ |
− |
− |
+ |
| y |
+ |
+ |
− |
− |
+ |
+ |
− |
− |
| z |
+ |
+ |
+ |
+ |
− |
− |
− |
− |
(i) (1, 2, 3) — x(+), y(+), z(+) → First Octant (I)
(ii) (4, −2, 3) — x(+), y(−), z(+) → Fourth Octant (IV)
(iii) (4, −2, −5) — x(+), y(−), z(−) → Eighth Octant (VIII)
(iv) (4, 2, −5) — x(+), y(+), z(−) → Fifth Octant (V)
(v) (−4, 2, −5) — x(−), y(+), z(−) → Sixth Octant (VI)
(vi) (−4, 2, 5) — x(−), y(+), z(+) → Second Octant (II)
(vii) (−3, −1, 6) — x(−), y(−), z(+) → Third Octant (III)
(viii) (−2, −4, −7) — x(−), y(−), z(−) → Seventh Octant (VII)
Q4. Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as ________.
Ans: In this plane, z = 0. The x-axis and y-axis taken together determine a plane known as the XY-plane.
(ii) The coordinates of points in the XY-plane are of the form ________.
Ans: In XY-plane, z = 0. Therefore, the coordinates of points in the XY-plane are of the form (x, y, 0).
(iii) Coordinate planes divide the space into _____ octants.
Ans: Coordinate planes divide the space into 8 octants.
FAQs – Class 11 Maths Chapter 11 Exercise 11.1 Introduction to Three Dimensional Geometry
Q1. What is the focus of Exercise 11.1? Exercise 11.1 introduces students to the three-dimensional coordinate system, including the three axes (x, y, z), three coordinate planes (XY, YZ, XZ), and the eight octants formed by these planes.
Q2. How do we identify the octant of a point in 3D geometry? The octant is identified by the signs of the x, y, and z coordinates of the point. For example, if x > 0, y > 0, z > 0, the point lies in the First Octant. If x > 0, y < 0, z > 0, it lies in the Fourth Octant.
Q3. What are the coordinates of a point on the y-axis? Any point on the y-axis has its x-coordinate and z-coordinate equal to zero. So its coordinates are of the form (0, y, 0).
Q4. What is the difference between 2D and 3D coordinate geometry? In 2D geometry, a point is represented by two coordinates (x, y). In 3D geometry, a point is represented by three coordinates (x, y, z), adding depth to the plane. The three coordinate planes (XY, YZ, XZ) replace the two axes of 2D geometry.
Q5. Why is Exercise 11.1 important for exams? This exercise forms the foundation of 3D geometry. Questions on identifying octants, coordinate axes, and coordinate planes are frequently asked in CBSE board exams. Understanding these basics is also essential for studying vectors and 3D geometry in Class 12.
Q6. How can students prepare effectively for Exercise 11.1? Students should memorize the octant sign table, understand the properties of each coordinate plane, and practice identifying the position of points in 3D space. Regular revision of the basic definitions will help in solving both direct and application-based problems.