Mathematics
Vector Projections and Orthogonal Decomposition
Quick fact
When you project vector a onto vector b, the orthogonal remainder (a - projb a) is always perpendicular to b. This simple idea is the foundation of the least squares method, which finds the best-fit line by making the errors exactly orthogonal to the trend.
Why this is interesting
Ever wondered why a shadow looks like it's stretched or squished? That's a vector projection at play—and it's also the secret behind splitting a force into parts that actually affect motion in a certain direction.