Q1. A point is on the x-axis. What are its y-coordinate and z-coordinates?

Solution

A point lying on the x-axis in three-dimensional space must satisfy the defining property of the x-axis. The x-axis consists of all points whose distances from both the y-axis and the z-axis are zero.

This means that for any point on the x-axis, the y-coordinate and the z-coordinate must be zero, while the x-coordinate may take any real value.

Thus, the general coordinates of a point on the x-axis can be written as

\[ (x, 0, 0) \]

Hence, the y-coordinate is \(0\) and the z-coordinate is also \(0\).

Q2. A point is in the XZ-plane. What can you say about its y-coordinate?

Solution

In three-dimensional Cartesian geometry, the XZ-plane consists of all points whose y-coordinate is zero. This is because the XZ-plane is formed by the x-axis and the z-axis, and every point lying on this plane has no displacement in the y-direction.

Hence, if a point lies in the XZ-plane, its coordinates are of the form (x, 0, z), where x and z are real numbers.

Mathematically, this condition can be expressed as

\[ \begin{aligned} y = 0 \end{aligned} \]

Therefore, the y-coordinate of a point lying in the XZ-plane is always zero.

Q3. 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

In three-dimensional Cartesian geometry, the octant in which a point lies depends on the signs of its x-, y-, and z-coordinates. Each combination of positive and negative signs determines a unique octant.

For the point \( (1, 2, 3) \), all three coordinates are positive. Hence, this point lies in the first octant.

For the point \( (4, -2, 3) \), the x-coordinate is positive, the y-coordinate is negative, and the z-coordinate is positive. Therefore, this point lies in the fourth octant.

For the point \( (4, -2, -5) \), the x-coordinate is positive while both the y- and z-coordinates are negative. Hence, this point lies in the eighth octant.

For the point \( (4, 2, -5) \), the x- and y-coordinates are positive and the z-coordinate is negative. Thus, the point lies in the fifth octant.

For the point \( (-4, 2, -5) \), the x-coordinate is negative, the y-coordinate is positive, and the z-coordinate is negative. Hence, the point lies in the sixth octant.

For the point \( (-4, 2, 5) \), the x-coordinate is negative while the y- and z-coordinates are positive. Therefore, this point lies in the second octant.

For the point \( (-3, -1, 6) \), the x- and y-coordinates are negative and the z-coordinate is positive. Hence, this point lies in the third octant.

For the point \( (-2, -4, -7) \), all three coordinates are negative. Therefore, this point lies in the seventh octant.

Q4. 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

The x-axis and y-axis together form a plane which is known as the XY-plane.

Any point lying in the XY-plane has no displacement along the z-direction. Therefore, the coordinates of such a point are of the form

\[ \begin{aligned} (x, y, 0) \end{aligned} \]

The three coordinate planes, namely the XY-plane, YZ-plane, and ZX-plane, together divide the three-dimensional space into eight distinct regions.

Hence, the coordinate planes divide the space into eight octants.