Lecture 3: Lines as vectors

Summary: Lines as Vectors

The lecture explains how to represent a geological line as a vector so it can be manipulated using linear algebra. Trend and plunge are useful for describing lines, but vector components make mathematical operations easier.

Coordinate Systems and Vector Basics

• A three-dimensional Cartesian coordinate system has three mutually perpendicular axes: (X_1), (X_2), and (X_3).
• The systems discussed are right-handed: curling the fingers of the right hand from the positive (X_2) direction toward the positive (X_3) direction points the thumb along (X_1).
• Two common geological coordinate conventions are:
  • East–North–Up (ENU): Common in topography and geophysics.
  • North–East–Down (NED): Common in structural geology.
• A vector has three components, written (v_1, v_2, v_3), or in index notation (v_i), where (i = 1, 2, 3).
• Its magnitude is the square root of the sum of its squared components:

[ |v| = \sqrt{v_1^2 + v_2^2 + v_3^2}. ]

• Dividing each component by the magnitude produces a unit vector, whose magnitude is 1. A hat above the vector symbol denotes a unit vector.

Direction Cosines

The components of a unit vector are the cosines of the angles it makes with the three coordinate axes. These are called direction cosines:

• (\cos\alpha): angle with the first axis
• (\cos\beta): angle with the second axis
• (\cos\gamma): angle with the third axis

In the NED system, a line plunging east has:

• A zero north component, because it is at (90^\circ) to the north axis.
• An east component equal to the cosine of its plunge.
• A down component equal to (\cos(90^\circ - \text{plunge})), or equivalently the sine of the plunge.

Converting Between Trend–Plunge and Vector Components

For a line with trend (T) measured from north and downward plunge (P), its NED unit-vector components can be written as:

[ (\cos\alpha, \cos\beta, \cos\gamma) = (\cos P \cos T,\ \cos P \sin T,\ \sin P). ]

The signs of the components indicate the line’s horizontal direction and whether it points up or down. For a downward-plunging line, the down component is positive. In the horizontal plane:

• Northeast: North and east components are positive.
• Southeast: North is negative and east is positive.
• Southwest: Both are negative.
• Northwest: North is positive and east is negative.

The conversion can also be reversed: trend and plunge can be calculated from the direction cosines. When finding trend with an arctangent, the result can be ambiguous between azimuths pointing in opposite directions. Use the component signs to choose the correct azimuth.

Suggested Practice

The lecturer recommends working through the relevant sections of the Computational Geosciences resource, then trying the accompanying exercise first by hand and again using the Python scripts introduced in class.

Speakers and Sources

• Speaker: One unnamed lecturer or narrator; no individual is identified in the subtitles.
• Sources mentioned: The Computational Geosciences resource and Python scripts introduced in class.
• Channel listed in the video context: University of Stavanger.