x-axis is the intersection of two planes (A) xy and xz (B) yz and zx (C) xy and yz (D) none of these

NCERT Exemplar · Q26

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Concept understanding — 3D Coordinate Octants

3D Coordinate Octants

Stand at the corner of a room where two walls meet the floor. That corner is the origin $O$ , and the three edges meeting there are the three coordinate axes:

the edge where the floor meets one wall is the x-axis,
the edge where the floor meets the other wall is the y-axis,
the vertical edge where the two walls meet is the z-axis.

These three axes are mutually perpendicular. Taken in pairs they determine three coordinate planes — the $xy$ -plane, the $yz$ -plane and the $zx$ -plane — and these planes slice all of space into eight regions. Each region is called an octant (from octo, meaning eight).

Why exactly eight

In a plane, the two axes make $4$ quadrants. Adding a third axis doubles this: each of the three coordinate planes splits space into two halves, so there are $2 \times 2 \times 2 = 2^3 = 8$ regions. Equivalently, an octant is fixed by choosing a sign — positive or negative — for each of $x$ , $y$ and $z$ , and there are $2^3 = 8$ such sign-triples $(\pm, \pm, \pm)$ .

The eight octants and their signs

(x, y, z)

The standard NCERT labelling of the signs is:

Octant	Sign of $x$	Sign of $y$	Sign of $z$	Example point
I	$+$	$+$	$+$	$(1, 2, 3)$
II	$-$	$+$	$+$	$(-1, 2, 3)$
III	$-$	$-$	$+$	$(-1, -2, 3)$
IV	$+$	$-$	$+$	$(1, -2, 3)$
V	$+$	$+$	$-$	$(1, 2, -3)$
VI	$-$	$+$	$-$	$(-1, 2, -3)$
VII	$-$	$-$	$-$	$(-1, -2, -3)$
VIII	$+$	$-$	$-$	$(1, -2, -3)$

Notice the pattern: octants I–IV all have $z > 0$ (above the $xy$ -plane) and V–VIII all have $z < 0$ (below it).

How to read a point's octant

Just look at the signs of its coordinates. For example, $(-3, 1, 2)$ has $x < 0$ , $y > 0$ , $z > 0$ , which is the sign pattern of octant II. The point $(-3, 1, -2)$ has $x < 0$ , $y > 0$ , $z < 0$ , placing it in octant VI. …

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The $x$ -axis is defined by points where both the $y$ -coordinate and $z$ -coordinate are zero. The $xy$ -plane has the equation $z=0$ , and the $xz$ -plane has the equation $y=0$ . Their intersection, where both $y=0$ and $z=0$ hold, precisely defines the $x$ -axis. Thus, option (A) is correct.

In three-dimensional Cartesian geometry, we use three mutually perpendicular axes, typically labeled $x$ , $y$ , and $z$ , which intersect at the origin $(0,0,0)$ . These axes define three fundamental coordinate planes. Understanding these planes and their equations is key to identifying their intersections.

The $xy$ -plane is the plane that contains the $x$ -axis and the $y$ -axis. Any point lying on this plane has its $z$ -coordinate equal to $0$ . Therefore, the equation of the $xy$ -plane is $z=0$ .
The $yz$ -plane is the plane that contains the $y$ -axis and the $z$ -axis. Any point lying on this plane has its $x$ -coordinate equal to $0$ . Therefore, the equation of the $yz$ -plane is $x=0$ .
The $xz$ -plane is the plane that contains the $x$ -axis and the $z$ -axis. Any point lying on this plane has its $y$ -coordinate equal to $0$ . Therefore, the equation of the $xz$ -plane is $y=0$ .

The intersection of two planes is the set of all points that lie on both planes simultaneously. Geometrically, the intersection of two non-parallel planes is a line. We need to find which pair of planes intersects to form the $x$ -axis.

Define the $x$ -axis: The $x$ -axis is the set of all points in 3D space where the $y$ -coordinate is $0$ and the $z$ -coordinate is $0$ . Such points have the form $(x, 0, 0)$ . Mathematically, the $x$ -axis is defined by the simultaneous equations:

y=0 \quad \text{and} \quad z=0

Examine Option (A): $xy$ and $xz$ planes
- The equation of the $xy$ -plane is $z=0$ .
- The equation of the $xz$ -plane is $y=0$ .
- The intersection of these two planes consists of all points $(x, y, z)$ that satisfy both $z=0$ and $y=0$ .
- This precisely matches the definition of the $x$ -axis. …

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Q.xxx-axis is the intersection of two planes (A) xyxyxy and xzxzxz (B) yzyzyz and zxzxzx (C) xyxyxy and yzyzyz (D) none of these

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Q. $x$ -axis is the intersection of two planes
(A) $xy$ and $xz$
(B) $yz$ and $zx$
(C) $xy$ and $yz$
(D) none of these

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