Where Mindful Learning Begins
Where Mindful Learning Begins
Complete step-by-step solutions for Exercise 11.1 of Introduction to Three Dimensional Geometry — coordinates on an axis and in a coordinate plane, and naming the octant of a point from the signs of its coordinates. Every question is solved in full, exam-ready detail as per the CBSE 2026-27 syllabus.
Any point on the x-axis is of the form (x,0,0), since it has zero perpendicular distance from both the y-axis direction and the z-axis direction.
The XZ-plane (also written ZX-plane) is the plane containing the x-axis and the z-axis. Every point in this plane has zero perpendicular distance from it measured along the y-axis.
The octant of a point is determined entirely by the signs of its x, y and z coordinates, according to the standard octant sign table:
| Octant | I | II | III | IV | V | VI | VII | VIII |
|---|---|---|---|---|---|---|---|---|
| x | + | − | − | + | + | − | − | + |
| y | + | + | − | − | + | + | − | − |
| z | + | + | + | + | − | − | − | − |
Matching the sign pattern of each point's coordinates to this table:
(1,2,3): signs (+,+,+) → octant I.
(4,-2,3): signs (+,−,+) → octant IV.
(4,-2,-5): signs (+,−,−) → octant VIII.
(4,2,-5): signs (+,+,−) → octant V.
(-4,2,-5): signs (−,+,−) → octant VI.
(-4,2,5): signs (−,+,+) → octant II.
(-3,-1,6): signs (−,−,+) → octant III.
(-2,-4,-7): signs (−,−,−) → octant VII.
(i) The plane containing both the x-axis and the y-axis is called the XY-plane.
(ii) Every point in the XY-plane has zero perpendicular distance from it measured along the z-axis, so its z-coordinate is always 0. A general point in the XY-plane is written as (x,y,0).
(iii) The three coordinate planes (XY, YZ and ZX) intersect each other along the coordinate axes and together divide the whole of space into 8 regions.
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