The functions $\sin$ and $\cos$ on $[-\pi, \pi]$, with the $L^2$ inner product $\ip{f}{g} = \int_{-\pi}^{\pi} f(x) g(x) \, dx$
Terms
| Term | This example | Why |
|---|---|---|
| orthogonal | ✓ | The functions $\sin$ and $\cos$ are orthogonal in the $L^2$ inner product on $[-\pi, \pi]$, because $\int_{-\pi}^{\pi} \sin(x) \cos(x) \, dx = 0$, since the integrand is odd. |
| perpendicular | ✗ | The functions $\sin$ and $\cos$ are not called perpendicular, because perpendicularity describes lines and planes in geometry, and functions are not lines. |
Distinctions using this example
- Right angle vs. perpendicular vs. orthogonal vs. normal: The functions $\sin$ and $\cos$ show that orthogonality applies to vectors that are not arrows in space, where “perpendicular” and “right angle” have no geometric meaning.