Geometry in Three-Dimensional Space

# Geometry in Three-Dimensional Space

We are about to extend our knowledge of Calculus, though before we do so, we will have to review some important material regarding three-dimensional space which we will denote by $\mathbb{R}^3$. Consider a point $P$ with coordinates $(x, y, z)$. Like in two-dimensional space, this ordered triple $(x, y, z)$ denotes the location of $P$ from the origin $(0, 0, 0)$. For example, the point $(1, 2, 3) \in \mathbb{R}^3$ represents a point $1$ along the positive $x$-axis, $2$ along the positive $y$-axis, and $3$ along the positive $z$-axis as depicted below.

What's also important to note is the labelling of the coordinate axes. We will use what is known as the right-handed orientation for defining the coordinate axes. A 3-dimensional coordinate system is said to be in this orientation if from the righthand your thumb, forefinger, and middle finger respectively represent the $x$-axis, $y$-axis, and $z$-axis. Furthermore, the space contained where $x, y, z ≥ 0$ is known as the first octant.

Now that we have established points and the coordinate system we will use on this site for $\mathbb{R}^3$, we can now look at other objects in three-dimensional space. Apart from points, we can represent various types of surfaces in $\mathbb{R}^3$ with equations. We will describe a handful of these various surfaces below.

Surface Image Examples Equations
Planes
The equation $x + y + z = 1$ represents a plane in $\mathbb{R}^3$, and contains the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Another example of a plane is the equation $x + y = 0$. Note that this sort of equation would normally represent a line in $\mathbb{R}^2$ but instead represents a plane in $\mathbb{R}^3$.
Spheres
The equation $x^2 + y^2 + z^2 = r^2$ represents a sphere with radius $r$ in $\mathbb{R}^3$. For example, $x^2 + y^2 + z^2 = 16$ represents a sphere with radius $4$.
Ellipsoids
The equation $\frac{x^2}{9} + \frac{y^2}{16} + \frac{z^2}{25} = 1$ represents an ellipsoid with semi-axes $3$, $4$, and $5$ along the $x$, $y$, and $z$ axes respectively.
Circular Cylinders
The equation $x^2 + y^2 = 25$ represents a circular cylinder with radius $5$ that is parallel to the $z$-axis. Another example of a circular cylinder can be generated by the equation $y^2 + z^2 = 100$ which has a radius of $10$ and is parallel to the $x$-axis.
Parabolic Cylinders
The equation $x = y^2$ represents a parabolic cylinder that is parallel to the $z$-axis.
Paraboloid
The equation $\frac{x^2}{9} + \frac{y^2}{16} = \frac{z}{25}$ represents a paraboloid that is parallel to the $z$-axis.

Apart from surfaces, sometimes equations in $\mathbb{R}^3$ depict two-dimensional or one-dimensional object. For example, certain equations in $\mathbb{R}^3$ can represent circles, lines, parabolas, which are two-dimensional objects, while other equations could represent just a single point which is a one-dimensional object. For example, the equation $x^2 + y^2 + z^2 = 0$ is satisfied by only the point $(x, y, z) = (0, 0, 0)$ and so $x^2 + y^2 + z^2 = 0$ represents the origin in $\mathbb{R}^3$.

Of course, there are many other possibilities that equations can represent in $\mathbb{R}^3$ that we'll see further on.