Index

Robotics / math

Unit vector

Any vector with magnitude of length 1 is considered a unit vector.

Normalize

scale down a vector to magnitude of 1, while preserving the direction

\[ \left\| v \right\| = \sqrt{x^{2}+y^{2}+z^{2}} \]

\[ u=\frac{v}{\left\| v \right\|} \]

import numpy as np

v = np.array([1,1])
magnitude = np.linalg.norm(v)
normalize_v = v / magnitude
print(normalize_v)

Dot product

stackoverflow godot vector math The dot product takes two vectors and returns a scalar:

var s = a.x*b.x + a.y*b.y + a.z*b.z

var s = a.dot(b)

• If the number is greater than zero, both are looking towards the same direction (the angle between them is < 90° degrees).
• If the number is less than zero, both are looking towards opposite direction (the angle between them is > 90° degrees).
• If the number is zero, vectors are shaped in L (the angle between them is 90° degrees).

dot product on unit vector

• If both vectors are facing towards the exact same direction (parallel to each other, angle between them is 0°), the resulting scalar is 1.
• If both vectors are facing towards the exact opposite direction (parallel to each other, but angle between them is 180°), the resulting scalar is -1.
• If their angle is 90°, then dot product is 0

This means that dot product between unit vectors is always between the range of 1 and -1

The dot product between two unit vectors is the cosine of the angle between those two vectors. So, to obtain the angle between two vectors.

angle_in_radians = acos( a.dot(b) )
# a and b are unit vector

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Cross product

3Blue1Brown - cross product

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Inverse matrix

Inverse 2*2 Matrix

Inverse of a 2*2 Matrix

\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

Check determinate

\[ \det(A) = ad - bc \]

Inverse (if determinate != 0)

\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

Inverse 3*3 matrix

Inverse of a 3*3 Matrix

import numpy as np

A = np.array([
    [1, 2, 3],
    [0, 1, 4],
    [5, 6, 0]
])

A_inv = np.linalg.inv(A)

print("A inverse:\n", A_inv)