Vector Algebra • structr

This tutorial introduces the basic mathematical operations of the structr package for analyzing orientation data.

library(structr)

Vector operations

Since the spherical and vector objects are easily convertible, they can be used for all sorts of vector operations, such as the magnitude (or length), the angle between vectors, dot product, cross product, projection and rotation.

Define some example vectors:

line1 <- Line(120, 50)
line2 <- Line(10, 30)

The vector length (or magnitude):

vector_length(line1)
#> [1] 1

Note

Orientation vectors are by definition unit vectors, i.e. their length is equal to 1.

The angle between two vectors:

angle(line1, line2)
#> [1] 78.89371

The dot product (or scalar product) of two vectors:

dotprod(line1, line2)
#> [1] 0.1926297

Intuitively, the dot product tells us how much two vectors point in the same direction. For unit vectors, it equals the cosine of the angle between them.

The cross product of two vectors:

crossprod(line1, line2)
#> Line object (n = 1):
#>   azimuth    plunge 
#> 258.66786  32.21399

This gives the vector that is perpendicular to the plane spanned by the two vectors. Geologically, the cross product of two lines is the pole to the plane containing both lines, and the cross product of two poles to planes is the line of intersection of the two planes.

The projection of a vector onto another vector:

project(line1, line2)
#> Line object (n = 1):
#> azimuth  plunge 
#>      10      30

Note

The projection of a vector onto a unit vector is parallel to that unit vector, with a length equal to their dot product (the cosine of the angle between them). As an orientation, it therefore points along the second vector.

The rotation of a vector about another vector (rotation axis) by a specified rotation angle:

rotate(line1, rotaxis = line2, rotangle = 45)
#> Line object (n = 1):
#>   azimuth    plunge 
#> 210.50391  70.01332

Right-hand-rule convention: Rotation is counterclockwise for positive angles.

Linear transformation transforms vectors using a transformation matrix (second-order tensor):

trans_mat <- matrix(runif(9), 3, 3)
transform_linear(line1, trans_mat)
#> Vector (Vec3) object (n = 1):
#>         x         y         z 
#> 0.4287424 1.0590220 0.4075062