Axes, planes and octants

What you will be able to do

• name the three axes, the three coordinate planes and the eight octants;
• read and write the coordinates of a point, and say which octant it is in;
• write the form of a point on an axis and of a point in a coordinate plane.

A fly in the corner of a room

A fly sits on the wall of a room. To say where it is, you pick the corner of the floor as your starting point. You say how far along the floor edge of one wall, how far along the floor edge of the other wall, and how high above the floor. Three numbers fix the place. That is why we say that space has three dimensions.

In the chapter on Straight Lines you used two perpendicular axes for a plane. In space we add a third axis.

The axes and the coordinate planes

Take three lines that meet at a point $O$, the origin, and that are perpendicular to each other. They are the $x$-axis, the $y$-axis and the $z$-axis. In our pictures the $x$-axis comes towards you, the $y$-axis runs to the right and the $z$-axis runs upwards.

The axes are in the right-handed order. Turn the $x$-axis towards the $y$-axis through a right angle. A screw turned this way moves along the positive $z$-axis. All books and exams use this order.

Any two axes lie in a coordinate plane:

• the $xy$-plane holds the $x$-axis and the $y$-axis. Every point on it has $z = 0$. It is the floor of our room.
• the $yz$-plane holds the $y$-axis and the $z$-axis. Every point on it has $x = 0$.
• the $zx$-plane holds the $z$-axis and the $x$-axis. Every point on it has $y = 0$.

The coordinates of a point

From a point $P$ drop perpendiculars to the three coordinate planes. Then

• $x$ is the signed distance of $P$ from the $yz$-plane,
• $y$ is the signed distance of $P$ from the $zx$-plane,
• $z$ is the signed distance of $P$ from the $xy$-plane.

We write $P(x, y, z)$. The signs follow the axes: the distance counts as positive on the side of the positive axis. The three perpendiculars are three edges of a box with $O$ and $P$ at opposite corners. The simulation below draws this box.

Points on an axis and in a plane

A point on an axis has two zero coordinates. A point in a coordinate plane has one zero coordinate.

The eight octants

The three coordinate planes cut space into eight octants, as two planes cut a sheet of paper into four quadrants. A point belongs to an octant only if none of its coordinates is zero. Then the signs of $x$, $y$, $z$ decide.

Octants I to IV lie above the $xy$-plane and follow the quadrants of the plane. Octant V is below octant I, octant VI below II, and so on. Some books list the octants by their axes, such as $OXYZ$ and $OX'YZ$, and some number them in another order. The signs always tell you where you are, so use them.

A worked example

In which octant is $P(-3, 4, -2)$?

1. Look at $z$. It is negative, so $P$ is below the $xy$-plane.
2. Look at $x$ and $y$. $x$ is negative and $y$ is positive, which is quadrant II of the plane.
3. Combine. Below octant II is octant VI. The signs $(-, +, -)$ match the table.

Practise with new numbers

Try it in Python

The function octant reads the signs of a point and gives its octant. A point with a zero coordinate gets a message instead.

Common mistakes

• Mixing up the order. The point $(2, 5, 7)$ has $x = 2$, $y = 5$, $z = 7$. Never swap the places.
• Forgetting the zero coordinates. The point on the $y$-axis at distance 3 is $(0, 3, 0)$, not $(3)$.
• Giving a point with a zero coordinate an octant. It lies on a plane, and the eight octants are the open regions between the planes.
• Using a different axis order. Keep $x$, $y$, $z$ right-handed. A left-handed set would give a mirror image of every answer.

Remember these

• The axes meet at right angles at the origin $O(0, 0, 0)$. The coordinate planes are $z = 0$, $x = 0$ and $y = 0$.
• $x$, $y$, $z$ are the signed distances from the $yz$-plane, the $zx$-plane and the $xy$-plane.
• A point on an axis has two zero coordinates. A point on a plane has one.
• Eight octants, decided by the three signs.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In which octant is the point $(-2, 5, 3)$?

2. In which octant is the point $(4, -3, -6)$?

3. Write the coordinates of the point on the $y$-axis that is 5 units from the origin on the negative side.

4. A point lies in the $zx$-plane. It is 4 units from the $yz$-plane on the negative side of $x$, and 7 units above the $xy$-plane. Write it.

5. Why does the point $(0, 2, -3)$ belong to none of the eight octants?

References

• NCERT, Mathematics Textbook for Class XI, the chapter Introduction to Three Dimensional Geometry, sections on the coordinate axes and planes.
• OpenStax, Calculus Volume 3, Chapter 2, section 2.2 on vectors in three dimensions.