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NCERT · Class 11 · Maths

Chapter 11: Introduction to Three Dimensional Geometry

Practice each question, then open the answer or worked solution.

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Exercise 11.1

4 questions
Q1Written answer

A point is on the x-axis. What are its y-coordinate and z-coordinates?

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Worked solution

Ans: We know that, the coordinates of any point on the $x$-axis will be as $\left( {x,0,0} \right)$.

Thus, if a point is on the $x$-axis, then its $y$-coordinate will be $0$ and $z$-coordinate will be 0.

Q2Written answer

A point is in the XZ-plane. What can you say about its y-coordinate?

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Worked solution

Ans: We know that, the coordinates of any point in the XZ-plane will be as $\left( {x,0,z} \right)$.

So, in the XZ- plane $y = 0$.

Thus, if a point is in the XZ- plane then the coordinate of $y$will always be 0.

Hence, its $y$-coordinate is 0.

Q3Written answer

Name the octants in which the following points lie:

$\left( {{\mathbf{1,2,3}}} \right)$

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Worked solution

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-


From the above table, the point $\left( {1,2,3} \right)$ lies in the first octant.

$\left( {{\mathbf{4, - 2,3}}} \right)$

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-


From the above table, the point $\left( {4, - 2,3} \right)$ lies in octant IV.

$\left( {{\mathbf{4, - 2, - 5}}} \right)$

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-


From the above table, the point $\left( {4, - 2, - 5} \right)$ lies in octant VIII.

$\left( {{\mathbf{4,2, - 5}}} \right)$

Ans: We have the following table,

From the above table, the point $\left( {4,2, - 5} \right)$ lies in octant V.

$\left( {{\mathbf{ - 4,2, - 5}}} \right)$

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-


From the above table, the point $\left( { - 4,2, - 5} \right)$ lies in octant VI.

$\left( {{\mathbf{ - 4,2,5}}} \right)$

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-

From the above table, the point $\left( { - 4,2,5} \right)$ lies in octant II.

$\left( {{\mathbf{ - 3, - 1,6}}} \right)$

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-

From the above table, the point $\left( { - 3, - 1,6} \right)$ lies in octant III.

$\left( {{\mathbf{ - 2, - 4, - 7}}} \right)$

Ans: We have the following table,

Octant Coordinates

I

II

III

IV

V

VI

VII

VIII

$x$ 

+

-

-

+

+

-

-

+

$y$ 

+

+

-

-

+

+

-

-

$z$ 

+

+

+

+

-

-

-

-

From the above table, the point $\left( { - 2, - 4, - 7} \right)$ lies in octant VII.

Q4Written answer

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.

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Worked solution

Ans:  In XY-plane, $z = 0$.

Hence, the $x$-axis and $y$-axis taken together determine a plane known as XY-plane.

Ans: In XY-plane, $z = 0$.

Let $x$-coordinate be $x$ and $y$-coordinate be $y$.

Hence, the coordinates of points in the XY-plane are of the form $\left( {x,y,0} \right)$.

Ans: Coordinate planes divide the space into 8 octants.