NCERT Class 11 Mathematics — Introduction to Three Dimensional Geometry, Exercise 11.1. All 4 questions solved.

Space is described by three mutually perpendicular axes $\mathrm{OX}$, $\mathrm{OY}$ and $\mathrm{OZ}$. Taken in pairs they determine three coordinate planes:

$$\mathrm{XY}\text{-plane}\ (z = 0), \qquad \mathrm{YZ}\text{-plane}\ (x = 0), \qquad \mathrm{ZX}\text{-plane}\ (y = 0)$$

These three planes cut space into eight octants, numbered by the sign pattern of $(x, y, z)$:

Key insight. The first four octants have $z > 0$ and the last four have $z < 0$, and within each group of four the $(x, y)$ signs run $(+,+)$, $(-,+)$, $(-,-)$, $(+,-)$ — exactly the four quadrants of the plane, in order. So an octant is found by reading the sign of $z$ to pick the group, then the $(x,y)$ signs to pick the member. Question 3 is eight applications of that rule.

Question 1

A point is on the $x$-axis. What are its $y$-coordinate and $z$-coordinate?

Solution. A point on the $x$-axis lies in both the $\mathrm{XY}$-plane and the $\mathrm{ZX}$-plane — it is where the two intersect. The $\mathrm{XY}$-plane forces $z = 0$ and the $\mathrm{ZX}$-plane forces $y = 0$.

So any point on the $x$-axis has the form $(x, 0, 0)$.

Question 2

A point is in the $\mathrm{XZ}$-plane. What can you say about its $y$-coordinate?

Solution. The $\mathrm{XZ}$-plane is spanned by the $x$- and $z$-axes, so moving within it never involves any displacement along $\mathrm{OY}$.

Every point of the $\mathrm{XZ}$-plane therefore has the form $(x, 0, z)$.

Question 3

Name the octants in which the following points lie:

$$(1,2,3),\ (4,-2,3),\ (4,-2,-5),\ (4,2,-5),\ (-4,2,-5),\ (-4,2,5),\ (-3,-1,6),\ (-2,-4,-7)$$

Solution. Read the signs of each triple against the table above.

Question 4

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 $\mathrm{XY}$-plane are of the form _______. (iii) Coordinate planes divide the space into _______ octants.

Solution.

(i) Two intersecting lines determine a unique plane, and the plane containing the $x$- and $y$-axes is by definition the $\mathrm{XY}$-plane.

(ii) Every point of the $\mathrm{XY}$-plane is at zero distance from it along $\mathrm{OZ}$, so its third coordinate vanishes: $(x, y, 0)$.

(iii) Each of the three coordinate planes splits space in two, and the three splits are independent, giving $2 \times 2 \times 2 = 8$ regions.

Common mistakes

• Question 1, giving only one zero. A point on an axis has two zero coordinates; a point in a plane has one. Distinguishing axis from plane is what the first two questions are testing.
• Question 2, setting $x = 0$ instead. The $\mathrm{XZ}$-plane contains the $x$- and $z$-axes, so it is $y$ that vanishes. The rule is that the missing letter is the coordinate that is zero.
• Question 3, numbering the octants by guesswork. The order is fixed by convention: octants I to IV lie above the $\mathrm{XY}$-plane and V to VIII below, each set running anticlockwise from the all-positive quadrant.
• Question 3, confusing octants V and VIII. Both have $z < 0$; they differ in the sign of $y$, which is $+$ for V and $-$ for VIII.
• Answering “four octants”. In two dimensions there are four quadrants; in three there are eight octants, because a third independent sign doubles the count.

Practise next

• Exercise 11.2 — the distance formula in space, and using it to test for collinearity, isosceles and right-angled triangles.
• Miscellaneous Exercise on Chapter 11 — parallelograms, medians and centroids in three dimensions.