Vector Projection
The problem, from first principles
"How much of vector a points in the same direction as vector b?" comes up constantly: it's the geometric idea behind a shadow, behind "how correlated are these two signals," and it's the exact operation 10-gaussian-elimination and 12-gram-schmidt both build on to remove one vector's overlap with another. This question implements it directly, before either of those questions needs it as a building block.
From theory to code
Implement project(a, b), returning the projection of a onto b — the component of a that points along b's direction — then orthogonal_component(a, b), the leftover part of a that's perpendicular to b. The signatures and docstrings are already in the editor.
Constraints
aandbare 1D NumPy arrays of the same length.bis never the zero vector (projecting onto a direction that doesn't exist is undefined).project(a, b) + orthogonal_component(a, b)must reconstructaexactly (up to floating-point precision).
Hints
Hint 1
The projection's length along b is (a · b) / (b · b) — a dot product ratio, not the dot product alone. Multiply that scalar by the vector b itself to get the actual projection vector.
Hint 2
Once you have project(a, b), the orthogonal component is just a minus it — whatever's left over after removing the along-b part must be everything perpendicular to b.
Keywords
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