Consider the two non-zero complex numbers `z` and `w` as vectors.
Which of the following expressions is the projection of `z` onto `w` ?
- `{text{Re} (zw)}/{|w|} w`
- `|z/w| w`
- `text{Re} (z/w) w`
- `{text{Re}(z)}/{|w|} w`
Show Worked Solution
`text{Let} \ \ z = a + i b \ \ =>\ underset~z = ((a),(b))`
♦♦♦ Mean mark 30%.
`text{Let} \ \ w = c + i d \ \ =>\ underset~w = ((c),(d))`
`underset~z * underset~w = ac + bd`
`|underset~w|^2 = c^2 + d^2`
`text{proj}_(underset~w) underset~z = (underset~z*underset~w)/|underset~w|^2 *underset~w= (ac+bd)/(c^2+d^2)*underset~w`
| `z/w` | `=(a+ib)/(c+id) xx (c-id)/(c-id)` | |
| `=(ac+bd + i(bc-ad))/(c^2+d^2)` |
`text{Re}(z/w) =(ac+bd)/(c^2+d^2)`
`:.\ text{proj}_(underset~w) underset~z = text{Re} (z/w) w`
`=> C`