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 \(x = {b \pm \sqrt{b^24ac} \over 2a}\) 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.