| vector | ruler | metric | |
|---|---|---|---|
| standard form | \( a \mult \vec{\standard}_{x} + b \mult \vec{\standard}_{y} \) | \( a \mult \bar{\standard}_{x} + b \mult \bar{\standard}_{y} \) | \( a \mult \hat{\standard}_{x} + b \mult \hat{\standard}_{y} \) |
| standards | \( \vec{\standard}_{x} \andSpaced \vec{\standard}_{y} \) | \( \bar{\standard}_{x} \andSpaced \bar{\standard}_{y} \) | \( \hat{\standard}_{x} \andSpaced \hat{\standard}_{y} \) |
| magnitude | length | unit circle measurement | |
| scaling | grows | shrinks | |
| operation | measurement | measurement | application |
| multiplication | anti-commutes | anti-commutes | commutes |
| poly- | length, area | slope, bend, warp | |
| differential | \( \bar{\diff} z \) | \( \hat{\diff} z \) |