Vector Algebra

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

library(structr)

Vector operations

Since the spherical or vector objects are easily convertible, they can be used for all sort 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

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.

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.

The projection of a vector on another vector:

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

Because the vectors are both unit vectors, the projected vector is equal to 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

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 
#>  0.44288029 -0.04205583  0.62796680

Tobias Stephan

2025-11-22

Vector operations