3D Geometry - Practice Questions & MCQ

As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal x-axis and the vertical y-axis. We can add a third dimension, the z-axis, which is perpendicular to both the x-axis and the y-axis. We call this system the three-dimensional rectangular coordinate system.

The system displayed follows the right-hand rule. If we take our right hand and align the fingers with the positive x-axis, then curl the fingers so they point in the direction of the positive y-axis, our thumb points in the direction of the positive z-axis. In this text, we always work with coordinate systems set up in accordance with the right-hand rule.

The three axes, X-axis, Y-axis and Z-axis are mutually perpendicular which defines three-dimensional (3D) coordinate system or space. Any point in this coordinate system has coordinates (x, y, z). Also, point in this system represents ordered triplets of set of cartesian product R x R x R Where, R is the set of real numbers.

If we take three axes, two at a time, then it will form three mutually perpendicular planes, XY-plane, YZ-plane and ZX-plane. This three plane divides space into eight regions called octants.

Coordinates of a Point in Space

Consider a point P(x, y, z) in three-dimensional system.

The x-coordinate of the point P is signed distance (OL) from the YZ-plane, i.e. a signed distance of P measured parallel to X-axis.

The y-coordinate of the point P is signed distance (ML) from the XZ-plane, i.e. a signed distance of P measured parallel to Y-axis.

The z-coordinate of the point P is signed distance (MP) from the XY-plane, i.e. a signed distance of P measured parallel to Z-axis.

The coordinates of the origin O are (0,0,0). The coordinates of any point on the x-axis will be as (x,0,0) and the coordinates of any point in the YZ-plane will be as (0, y, z).

The sign of the coordinates of a point determines the octant in which the point lies. The following table shows the signs of the coordinates in eight octants.

Distance between Two Points:

Let P(x₁ , y₁ , z₁ ) and Q ( x₂ , y₂ , z₂ ) be two points in a three-dimensional system of rectangular axes OX, OY and OZ.

Since, PAQ is a right-angled triangle with ∠PAQ as a right angle,

Using the Pythagorean theorem
$
\mathrm{PQ}^2=\mathrm{PA}^2+\mathrm{AQ}^2
$

Also, triangle ANQ is a right angle triangle with $\angle A N Q$ aright angle.
$\therefore$
$
\mathrm{AQ}^2=\mathrm{AN}^2+\mathrm{QN}^2
$
from (i) and (ii)
$
\mathrm{PQ}^2=\mathrm{PA}^2+\mathrm{AN}^2+\mathrm{QN}^2
$

Now,
$
\mathrm{PA}=y_2-y_1, \mathrm{AN}=x_2-x_1 \text { and } \mathrm{NQ}=z_2-z_1
$

Hence,
$
\begin{aligned}
& \mathrm{PQ}^2=\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2 \\
\therefore & \mathrm{PQ}=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}
\end{aligned}
$

This gives us the distance between two points P(x₁ , y₁ , z₁ ) and Q ( x₂ , y₂ , z₂ )

Distance of a point from the Axes and Origin

Distance of a point from $x$-axis is $\sqrt{\mathrm{y}^2+\mathrm{z}^2}$.
Distance of a point from $y-a x i s$ is $\sqrt{\mathrm{x}^2+\mathrm{z}^2}$.
Distance of a point from $z-a x i s$ is $\sqrt{\mathrm{y}^2+\mathrm{x}^2}$.

Distance from Origin
If $x_2=y_2=z_2=0$, i.e., point $Q$ is origin $O$, then $\mathrm{OP}=\sqrt{x_1^2+y_1^2+z_1^2}$

which gives the distance between the origin O and any point P(x₁, y₁, z₁)