Coordinate Axes, Planes and Octants

Three Axes, Three Planes, Eight Octants

Locating a point in a plane takes two numbers; locating a fan-tip in a room takes three — the distances from two adjacent walls and from the floor. That is the whole idea of 3D coordinates.

Take three mutually perpendicular planes meeting at a point $O$. They intersect along three mutually perpendicular lines $X'OX$, $Y'OY$, $Z'OZ$ — the $x$-, $y$- and $z$-axes — forming the rectangular coordinate system with origin $O$.

Key Point (Coordinate planes): each pair of axes spans a plane — the $XY$-plane, $YZ$-plane and $ZX$-plane. With the $XY$-plane horizontal, distances up along $OZ$ are positive, down are negative; similarly each axis carries its positive and negative direction.

The three planes slice space into eight octants, numbered I through VIII. Signs identify them:

Reading the table: $(-3, 1, 2)$ has signs $(-, +, +)$ — octant II; $(-3, 1, -2)$ has signs $(-, +, -)$ — octant VI. The pattern: octants I-IV have $z > 0$ and cycle through the four 2D-quadrant sign patterns of $(x, y)$; octants V-VIII repeat them with $z < 0$.

Coordinates of a Point in Space

Key Point (The triplet): given $P$ in space, drop a perpendicular $PM$ to the $XY$-plane, then a perpendicular $ML$ to the $x$-axis. With $OL = x$, $LM = y$, $MP = z$, the point is $P(x, y, z)$. Equivalently, $x$, $y$, $z$ are the perpendicular distances of $P$ from the $YZ$-, $ZX$- and $XY$-planes respectively.

This sets up a one-to-one correspondence between points of space and ordered triplets $(x, y, z)$: every point gets exactly one triplet, and every triplet locates exactly one point (fix $L$ on the $x$-axis, walk to $M$ in the $XY$-plane, rise to $P$).

Special positions worth memorising:

The origin is $(0, 0, 0)$. A point on the $x$-axis looks like $(x, 0, 0)$; on the $y$-axis $(0, y, 0)$; on the $z$-axis $(0, 0, z)$. A point in the $XY$-plane looks like $(x, y, 0)$; in the $YZ$-plane $(0, y, z)$; in the $ZX$-plane $(x, 0, z)$.

Worked check: if $P = (2, 4, 5)$ and $F$ is the foot of the perpendicular from $P$ to the $ZX$-plane, then the $y$-distance collapses to zero: $F = (2, 0, 5)$.

[Board Tip] "A point is in the $XZ$-plane — what is its $y$-coordinate?". Answer in one word: zero. Lying in a coordinate plane kills exactly the coordinate measured away from that plane — no computation needed.

Solved Examples

Example 1: Collapsing one coordinate

$P$ is $(2, 4, 5)$. Find the coordinates of $F$, the foot of the perpendicular from $P$ to the $ZX$-plane.

Solution:

Step 1 — Identify what reaching the plane changes. The $ZX$-plane is the set $y = 0$; the perpendicular from $P$ travels purely in the $y$-direction, so only the $y$-coordinate changes.

Step 2 — Set it to zero. $F = (2, 0, 5)$.

Takeaway: Dropping a perpendicular to a coordinate plane zeroes exactly one coordinate — the one measured away from that plane.

Example 2: Naming octants

Find the octants in which $(-3, 1, 2)$ and $(-3, 1, -2)$ lie.

Solution:

Step 1 — Read the signs. $(-3, 1, 2)$ has sign pattern $(-, +, +)$; $(-3, 1, -2)$ has $(-, +, -)$.

Step 2 — Look up the octant table. $(-, +, +)$ is octant II; $(-, +, -)$ is octant VI.

Takeaway: Octants I-IV have $z > 0$ and cycle the 2D quadrant signs of $(x, y)$; V-VIII repeat them with $z < 0$ — the same point with $z$ negated jumps by exactly four.

Example 3: Points on an axis, points in a plane

(i) A point lies on the $x$-axis: what are its $y$- and $z$-coordinates? (ii) A point lies in the $XZ$-plane: what is its $y$-coordinate?

Solution:

Step 1 — (i) An axis point. The $x$-axis is the intersection of the $XY$- and $ZX$-planes, so distances from both are zero: $y = 0$ AND $z = 0$ — the point is $(x, 0, 0)$.

Step 2 — (ii) A plane point. Lying in the $XZ$-plane means zero distance from it, and that distance is $|y|$: $y = 0$, form $(x, 0, z)$.

Takeaway: Membership of an axis kills two coordinates; membership of a coordinate plane kills one.

Example 4: An octant drill

Name the octants of $(1, 2, 3)$, $(4, -2, 3)$, $(4, -2, -5)$, $(4, 2, -5)$, $(-4, 2, -5)$, $(-4, 2, 5)$, $(-3, -1, 6)$, $(-2, -4, -7)$.

Solution:

Step 1 — Sort by the sign of $z$. With $z > 0$: $(1,2,3)$, $(4,-2,3)$, $(-4,2,5)$, $(-3,-1,6)$ land in octants I-IV; with $z < 0$: the other four land in V-VIII.

Step 2 — Use the $(x, y)$ quadrant within each half. $(1,2,3)$: $(+,+)$ → I. $(4,-2,3)$: $(+,-)$ → IV. $(-4,2,5)$: $(-,+)$ → II. $(-3,-1,6)$: $(-,-)$ → III.

Step 3 — Repeat below the $XY$-plane. $(4,2,-5)$: $(+,+)$ → V. $(-4,2,-5)$: $(-,+)$ → VI. $(-2,-4,-7)$: $(-,-)$ → VII. $(4,-2,-5)$: $(+,-)$ → VIII.

Takeaway: Decide $z$ first, then read the $(x, y)$ pair like a 2D quadrant — the table becomes automatic.

Example 5: Fill in the blanks

(i) The $x$-axis and $y$-axis taken together determine a plane known as . (ii) The coordinates of points in the $XY$-plane are of the form . (iii) Coordinate planes divide the space into __ octants.

Solution:

Step 1 — (i). Two intersecting lines span exactly one plane: the $XY$-plane.

Step 2 — (ii). Height above that plane is $z$, and in the plane $z = 0$: form $(x, y, 0)$.

Step 3 — (iii). Each coordinate independently positive or negative: $2^3 =$ eight octants.

Takeaway: All three blanks are one fact seen three ways — the coordinate planes carve space by signs.

Example 6: Signs to octant, octant to signs

(i) A point has $x < 0$, $y < 0$, $z > 0$: which octant? (ii) Write a sample point in octant VII.

Solution:

Step 1 — (i) Signs to octant. $(-, -, +)$: third quadrant pattern in $(x,y)$ with $z > 0$ — octant III.

Step 2 — (ii) Octant to signs. Octant VII is the $z < 0$ partner of III: signs $(-, -, -)$, e.g. $(-1, -2, -3)$.

Takeaway: The table works in both directions — memorise the four upper octants and negate $z$ for the lower four.

Example 7: Feet of perpendiculars to all three planes

For $P(3, -4, 5)$, write the feet of the perpendiculars from $P$ to the $XY$-, $YZ$- and $ZX$-planes, and the distances to each plane.

Solution:

Step 1 — Zero one coordinate per plane. Foot on $XY$ ($z=0$): $(3, -4, 0)$; on $YZ$ ($x=0$): $(0, -4, 5)$; on $ZX$ ($y=0$): $(3, 0, 5)$.

Step 2 — Distances are the dropped coordinates. From $XY$: $|z| = 5$; from $YZ$: $|x| = 3$; from $ZX$: $|y| = 4$.

Step 3 — Check one. From $(3,-4,5)$ to $(3,-4,0)$: $\sqrt{0+0+25} = 5$ ✓.

Takeaway: $|x|, |y|, |z|$ ARE the three plane-distances — that is the geometric meaning of the coordinates.

Example 8: Reflections in the coordinate planes

Write the image of $P(2, -3, 5)$ in (i) the $XY$-plane, (ii) the $YZ$-plane, (iii) the origin.

Solution:

Step 1 — (i) Mirror in $XY$. The mirror flips the coordinate perpendicular to it: $z \to -z$ gives $(2, -3, -5)$.

Step 2 — (ii) Mirror in $YZ$. $x \to -x$ gives $(-2, -3, 5)$.

Step 3 — (iii) Point reflection in $O$. All three flip: $(-2, 3, -5)$.

Takeaway: Plane mirrors flip ONE sign; the origin flips all three; an axis reflection (compose two planes) flips two.