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Lesson Directive // Orbital TransformationsREF_CORE

Coordinate Systems

v⃗′\vec{v}'==R(θ)R(\theta)⋅\cdotv⃗\vec{v}

Hover over a variable in the formula above, or see glossary below:

v⃗′\vec{v}'
Rotated Vector
Coordinates
R(θ)R(\theta)
Rotation Matrix
Matrix
v⃗\vec{v}
Original Vector
Coordinates

Imagine looking at a Vector arrow pointing 'Up'. Now tilt your head sideways. The arrow is still pointing to the same physical place, but to your eyes, it's now pointing 'Right'. Spacecraft do this constantly, rotating their frames of reference from pointing at the Sun, to pointing at Earth.

INSIGHT: To translate data between these systems without physically moving, we mathematically rotate the vectors.

The Rotation Matrix

A rotation matrix is like a mathematical machine. You feed it your original X and Y coordinates, and it multiplies them by Trigonometric functions (sines and cosines) to spit out the brand new rotated coordinates.

INSIGHT: Matrix multiplication smoothly combines the old coordinates to form the new ones.

Preserving Magnitude

A pure rotation matrix changes the direction of a vector but preserves its magnitude (length). The spacecraft's speed or distance remains the exact same, we've just pivoted the camera.

INSIGHT: Rotation matrices are orthogonal, meaning they do not stretch or squish the vector.