NCERT Class 11 Maths Introduction to Three Dimensional Geometry

Chapter 12 Ex.12.1 Question 1

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

Solution

If a point is on the $$x -$$axis, then the coordinates of $$y$$ and $$z$$ are $$0.$$

So, the point is $$\left( x,0,0 \right)$$.

Chapter 12 Ex.12.1 Question 2

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

Solution

If a point is in $$XZ -$$plane, then its $$y -$$coordinate is $$0.$$

Chapter 12 Ex.12.1 Question 3

Name the octants in which the following points lie:

\begin{align}&\left( {1,2,3} \right),\;\left( {4, - 2,3} \right),\;\left( {4, - 2, - 5} \right),\\&\left( {4,2, - 5} \right),\;\left( { - 4,2, - 5} \right),\;\left( { - 4,2,5} \right),\;\\&\left( { - 3, - 1,6} \right),\left( {2, - 4, - 7} \right). \end{align}

Solution

Here is the table which represents the octants:

 Octants I II III IV V VI VII VIII $$x$$ $$+$$ $$-$$ $$-$$ $$+$$ $$+$$ $$-$$ $$-$$ $$+$$ $$y$$ $$+$$ $$+$$ $$-$$ $$-$$ $$+$$ $$+$$ $$-$$ $$-$$ $$z$$ $$+$$ $$+$$ $$+$$ $$+$$ $$-$$ $$-$$ $$-$$ $$-$$

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

Here $$x$$ is positive, $$y$$ is positive, and $$z$$ is positive.

So, it lies in $$I$$ octant.

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

Here $$x$$ is positive, $$y$$ is negative, and $$z$$ is positive.

So, it lies in IV octant.

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

Here $$x$$ is positive, $$y$$ is negative, and $$z$$ is negative.

So, it lies in VIII octant.

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

Here $$x$$ is positive, $$y$$ is positive, and $$z$$ is negative.

So, it lies in V octant.

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

Here $$x$$ is negative, $$y$$ is positive, and $$z$$ is negative.

So, it lies in VI octant.

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

Here $$x$$ is negative, $$y$$ is positive, and $$z$$ is positive.

So, it lies in II octant.

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

Here $$x$$ is negative, $$y$$ is negative, and $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$ is positive.

So, it lies in III octant.

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

Here $$x$$ is positive, $$y$$ is negative, and $$z$$ is negative.

So, it lies in VIII octant.

Chapter 12 Ex.12.1 Question 4

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 $$\rm{XY}$$-plane are of the form _____________.

(iii) Coordinate planes divide the space into _____________octants..

Solution

(i) The x-axis and y-axis taken together determine a plane known as XY plane.

(ii) The coordinates of points in the $$\rm{XY}$$-plane are of the form (x, y, 0).

(iii) Coordinate planes divide the space into eight octants.