Question

In which Octant does (-1,-5,-7) lie?
\[\begin{align}
  & A.I \\
 & B.VII \\
 & C.V \\
 & D.III \\
\end{align}\]

Answer 1

Hint: In this question, we need to locate the octant in which the point (-1,-5,-7) lies. For this, we will understand the three-dimensional coordinate system and then learn the type of points which lie in particular octant. Then we will compare the type with the given point and find our required octant.

Complete step by step answer:
Let us first learn about the octants in three-dimensional geometry.
When three axis, x axis, y axis and z axis cut each other perpendicularly in three dimensions, light octants are formed. Every octant has specific types of points such as $ \left( -x,-y,z \right),\left( x,y,-z \right)\text{ and }\left( x,y,z \right) $ etc. Let us learn what type of points are in each octant.
In the first octant, the x coordinate is positive, the y coordinate is positive and the z coordinate is positive. So octant I have points of the form (x,y,z).
In the second octant, x is negative, y is positive and z is positive. So octant II has points of the form (-x,y,z).
In the third octant, x is negative, y is negative and z is positive. So octant III has points of the form (-x,-y,z).
In the fourth octant, x is positive, y is negative and z is positive. So octant IV has points of the form (x,-y,z).
In the fifth octant, x is positive, y is positive and z is negative. So octant V has points of the form (x,y,-z).
In the sixth octant, x is negative, y is positive and z is negative. So, octant VI has points of the form (-x,y,-z).
In the seventh octant, x is negative, y is negative and z is negative. So, octant VII has points of the form (-x,-y,-z).
In the eighth octant, x is positive, y is negative and z is negative. So, octant VIII has points of the form (x,-y,-z).
Our point is (-1,-5,-7) which is of the form (-x,-y,-z). So, point (-1,-5,-7) lies in VIII octant.
Hence option B is the correct answer.

Note:
 Students should know all the points of the form in different octant. Take care of the signs for every form of point. Students should note that there are eight octants each having different types of points.