complex vectors ex 2.3 (1 Viewer)

[.ben]

4. On an Argand diagram the point P and Q represent z and z+iz respectively. Show that OPQ is a right angled triangle.

5. On an Argand diagram the point P and Q represent z1 and z2 respectively. OPQ is an equilateral triangle. Show that z1²+z2²=z1z2

 

[pLuvia]

4

Let
z = x+yi
iz = -y+xi

--->OR = iz

^POR = pi/2 |z| = |iz|
Triangle OPQ is a right angled

5

Let z₁ = z₂ cis pi/3
z²₁ = z²₂ cis 2pi/3

LHS = z²₁ + z²₂
= z²₂cis 2pi/3 + z²₂
= z²₂(cis 2pi/3 +1)
= z²₂(1 + cos 2pi/3 + isin 2pi/3)
= z²₂(1/2 + i(root 3)/2)

RHS = z₁. z₂
= z₂ cis pi/3 . z₂
= z²₂ cis pi/3
= z²₂(1/2 + i(root3)/2)
= LHS as req'd

 

[.ben]

cool thanks man!