Dated: 30-10-2024

Elements of Three Dimensional Geometry

Distance In 3D

Imagine we have 2 points in space, \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\), then the distance formula for these points will be

\[\lvert \overline{PQ} \rvert = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Midpoint Of 2 Points

If \(R\) is the midpoint between \(\overline{PQ}\) then it can be defined as

\[R\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)\]

Directional Angles

The angles \(\alpha\), \(\beta\) and \(\gamma\) are the angles between the line¹(or you can say its shadows on the \(xy\), \(yz\) and \(xz\) planes²) and the \(x\), \(y\) and \(z\) axes respectively.

\(\gamma\) can work in both \(xz\) and \(yz\) planes²

Directional Ratios

The cosines of directional angles are called directional cosines.
And any multiple of directional cosines are called directional ratios.

Let us talk about the shadow on the \(xy\) plane.²

\[\cos \alpha = \frac x r = \frac{x}{\sqrt{x^2 + y^2 + z^2}}\]

\[\cos \beta = \frac y r = \frac{y}{\sqrt{x^2 + y^2 + z^2}}\]

\[\cos \gamma = \frac z r = \frac{z}{\sqrt{x^2 + y^2 + z^2}}\]

\[\therefore \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]

For a line,¹ joining \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\), the directional cosines are

\[\cos \alpha = \frac{x_2 - x_1}{\lvert \overline{PQ} \rvert}\]

\[\cos \beta = \frac{y_2 - y_1}{\lvert \overline{PQ} \rvert}\]

\[\cos \gamma = \frac{z_2 - z_1}{\lvert \overline{PQ} \rvert}\]

Intersection of 2 Non-parallel planes²

The intersection between 2 non-parallel planes² creates a line.¹
Their intersection gives us simultaneous equations which are called non symmetric form of equations of a straight line.¹

Region	Description	Equation
xy plane	consists of points of form \((x, y, 0)\)	\(z = 0\)
yz plane	consists of points of form \((0, y, z)\)	\(x = 0\)
xz plane	consists of points of form \((x, 0, z)\)	\(y = 0\)
x axis	consists of points of form \((x, 0, 0)\)	\(y = 0, z = 0\)
y axis	consists of points of form \((0, y, 0)\)	\(x = 0, z = 0\)
z axis	consists of points of form \((0, 0, z)\)	\(x = 0, y = 0\)

General Equation of plane²

\[ax + by + cz + d = 0\]

where \(a\), \(b\), \(c\) and \(d \in \mathbb{R}\) .

Equation of Sphere

\[\sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2} = r\]

Where \(r\) is the radius of the sphere centered at \(O(x_0, y_0, z_0)\).

Equation of a Right Circular Cone

\[\phi = \frac \pi 4\]

\[z = \sqrt{x^2 + y^2}\]

Here \(\phi\) is \(\gamma\).

Elliptic Cylinder

From the equation of ellipse

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = r^2\]

We can also introduce a 3rd axis \(z = c\) to define the height,

Reference

Read more about notations and symbols.

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1. Read more about lines. ↩↩↩↩

2. Read more about planes. ↩↩↩↩↩↩