Question

How many of these are correctly matched ?

Coordinates of a point Name of the octant

1. $(1,2,3)P.I$

2. $(4,-2,3)Q.IV$

3. $(4,-2,-5)R.VIII$

4. $(4,2,-5)S.V$

5. $(-4,2,-5)T.VI$

6. $(-4,2,5)U.II$

7. $(-3,-1,6)V.III$

8. $(2,-4,-7)W.VIII$

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Answer 1

The sign of the coordinates of a point determine the octant in which the points lies.

The following table shows the signs of the coordinates in eight octants.

$\begin{array}{ccccccccc}\frac{octant}{coordinates}& \text{I}& \text{II}& \text{III}& \text{IV}& \text{V}& \text{VI}& \text{VII}& \text{VIII}\\ x& \text{+}& \text{-}& \text{-}& \text{+}& \text{+}& \text{-}& \text{-}& \text{+}\\ y& \text{+}& \text{+}& \text{-}& \text{-}& \text{+}& \text{+}& \text{-}& \text{-}\\ z& \text{+}& \text{+}& \text{+}& \text{+}& \text{-}& \text{-}& \text{-}& \text{-}\end{array}$

Now we can observe the given coordinates and find the correct octant for those coordinates.

Point $(1,2,3)\equiv (+,+,+)$ lies in I octant

Point $(4,-2,3)\equiv (+,-,+)$ lies in IV octant

Point $(4,-2,-5)\equiv (+,-,-)$ lies in VIII octant

Point $(4,2,-5)\equiv (+,+,-)$ lies in V octant

Point $(-4,2,-5)\equiv (-,+,-)$ lies in VI octant

Point $(-4,2,5)\equiv (-,+,+)$ lies in II octant

Point $(-3,-1,6)\equiv (-,-,+)$ lies in III octant

Point $(2,-4,-7)\equiv (+,-,-)$ lies in VIII octant

All of them are matched correct.