smc-7430-70-Vectors

Complex Numbers, EXT2 N2 2021 HSC 10 MC

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`

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`C`

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`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`

Complex Numbers, EXT2 N2 2017 HSC 13e

The points `A, B, C` and `D` on the Argand diagram represent the complex numbers `a, b, c` and `d`respectively. The points form a square as shown on the diagram.

By using vectors, or otherwise, show that `c = (1 + i) d-ia`. (2 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`

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♦ Mean mark 49%.

`c-d`	`= i (d-a)\ \ \ text{(rotation of}\ pi/2 text{)}`
`:. c `	`= d + id-ia`
	`= (1 + i) d-ia\ text(… as required.)`

Complex Numbers, EXT2 N2 2012 HSC 12d

On the Argand diagram the points `A_1` and `A_2` correspond to the distinct complex numbers `u_1` and `u_2` respectively. Let `P` be a point corresponding to a third complex number `z`.

Points `B_1` and `B_2` are positioned so that `ΔA_1PB_1` and `ΔA_2B_2P`, labelled in an anti-clockwise direction, are right-angled and isosceles with right angles at `A_1` and `A_2`, respectively. The complex numbers `w_1` and `w_2` correspond to `B_1` and `B_2`, respectively.

Explain why `w_1 = u_1 + i(z-u_1)`. (1 mark)

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Find the locus of the midpoint of `B_1B_2` as `P` varies. (2 marks)

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a. `text(See Worked Solutions.)`

b. `(u_1 + u_2)/2 + (u_2-u_1)/2 i`

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a.	`vec (A_1P)`	`= z-u_1`
	`vec (A_1B_1)`	`= w_1-u_1`

`B_1A_1 ⊥ A_1P\ text(and)\ |vec (A_1P)| = |vec (A_1B_1)|`

`vec (A_1B_1)\ text(is an anticlockwise rotation of)\ vec (A_1P)\ text(through)\ 90^@`

`w_1-u_1 = i(z −u_1)\ \ =>\ \ w_1 = u_1+ i(z −u_1)`

b.	`vec (A_2B_2)`	`= w_2-u_2`
	`vec (A_2P)`	`= z-u_2`

`A_2B_2 ⊥ A_2P\ text(and)\ |vec (A_2B_2)| = |vec (A_2P)|`

`vec (A_2P)\ text(is an anticlockwise rotation of)\ vec (A_2B_2)\ text(through)\ 90^@`

`z-u_2`	`= i(w_2 −u_2)`
`iw_2`	`= z-u_2 + iu_2`
`−w_2`	`= iz-iu_2-u_2`
`:. w_2`	`= u_2 + i(u_2-z)`

`:.\ text(The midpoint of)\ B_1B_2\ text(is)\ (w_1 + w_2)/2`

`= 1/2[u_1 + i(zvu_1) + u_2 + i(u_2-z)]`

`= 1/2[u_1 + u_2 + i(u_2-u_1)]`

`= (u_1 + u_2)/2 + (u_2-u_1)/2 i\ \ \ \ text{(which is a fixed point)}`