ALGEBRA OF COMPLEX NUMBERS (REVISION)

\[1.\ Find\ the\ modulus\ and\ amplitude\ of\ \frac{1\ +\ 3\sqrt{3}\ i}{\sqrt{3}\ +\ 2\ i}\ \hspace{18cm}\]
\[\color {black}{Solution:} \hspace{20cm}\]
\[Let\ z = \frac{1\ +\ 3\sqrt{3}\ i}{\sqrt{3}\ +\ 2\ i}\ \hspace{18cm}\]
\[= \frac{1\ +\ 3\sqrt{3}\ i}{\sqrt{3}\ +\ 2\ i}\ ×\ \frac{\sqrt{3}\ -\ 2\ i}{\sqrt{3}\ -\ 2\ i}\ \hspace{10cm}\]
\[ = \frac{\sqrt{3}\ +\ \sqrt{3}(3\sqrt{3}\ i)\ -\ 2\ i\ -\ 6\ \sqrt{3}\ (i^2)}{(\sqrt{3}\ +\ 2\ i)\ (\sqrt{3}\ -\ 2\ i) }\ \hspace{6cm}\]
\[ = \frac{\sqrt{3}\ +\ 9\ i\ -\ 2\ i + \ 6\ \sqrt{3}}{(\sqrt{3})^2 + (2)^2}\ \hspace{3cm}\ \because [(a +ib)(a – ib)\ =\ (a)^2 + (b)^2]\]
\[ = \frac{7\ \sqrt{3}\ +\ 7\ i}{3+ 4}\ \hspace{4cm}\ \because [i^2\ =\ -1]\]
\[ = \frac{7\ \sqrt{3}\ +\ 7\ i}{7}\ \hspace{10cm}\]
\[ Z = \frac{7\ \sqrt{3}}{7}\ +\ \frac{7\ i}{7}\ \hspace{15cm}\]
\[ z\ =\ \sqrt{3}\ + \ i\ = a\ + ib\ \hspace{14cm}\]
\[a\ =\ \sqrt{3},\ b\ = \ 1\ \hspace{15cm}\]
\[\color {brown} {T0\ find\ modulus}:\ \hspace{18cm}\]
\[|z| = \sqrt{a^2 + b^2}\ \hspace{12cm}\]
\[ = \sqrt{(\sqrt{3})^2 + {1}^2}\ \hspace{12cm}\]
\[ = \sqrt{3\ +\ 1}\ =\ \sqrt{4}\ =\ 2\ \hspace{12cm}\]
\[|z|\ =\ 2\ \hspace{15cm}\]
\[\color {brown} {To\ find\ amplitude}:\ \hspace{18cm}\]
\[θ = tan^{-1} (\frac{b}{a})\ =\ tan^{1} (\frac{1}{\sqrt{3}})\ \hspace{11cm}\]
\[θ = 30^0\ \hspace{15cm}\]
\[2.\ Prove\ that\ the\ complex\ numbers\ 3\ +\ 4\ i,\ 9\ +\ 8\ i,\ 5\ +\ 2\ i\ and\ -\ 1\ -\ 2\ i\ form\ a\ rhombus\ \hspace{10cm}\]\[in\ the\ argand\ diagram\ \hspace{10cm}\]
\[\color {black}{Solution:} \hspace{20cm}\]
\[Let\ A = 3\ +\ 4\ i\ = (3, 4)\ \hspace{15cm}\]
\[ B = 9\ +\ 8\ i\ = (9, 8)\ \hspace{13cm}\]
\[ C = 5\ +\ 2\ i\ = (5, 2)\ \hspace{13cm}\]
\[ D = -\ 1\ -\ 2\ i\ = (-\ 1, 2)\ \hspace{13cm}\]
\[AB\ =\ \sqrt{ (x_1\ – x_2)^2 + (y_1\ – y_2)^2 }\ \hspace{8cm}\]
\[=\ \sqrt{ (3\ -\ 9)^2 + (4\ -\ 8)^2 }\ \hspace{8cm}\]
\[ =\ \sqrt{ ((-\ 6)^2 + (-\ 4)^2 )}\ \hspace{8cm}\]
\[ =\ \sqrt{ (36\ +\ 16 )}\ \hspace{8cm}\]
\[AB = \sqrt{52}\ \hspace{8cm}\]
\[BC\ =\ \sqrt{ (x_1\ – x_2)^2 + (y_1\ – y_2)^2 }\ \hspace{8cm}\]
\[=\ \sqrt{ (9\ -\ 5)^2 + (8\ -\ 2)^2 }\ \hspace{8cm}\]
\[ =\ \sqrt{ ((4)^2 + (6)^2 )}\ \hspace{8cm}\]
\[ =\ \sqrt{ (16\ +\ 36 )}\ \hspace{8cm}\]
\[BC = \sqrt{52}\ \hspace{8cm}\]
\[CD\ =\ \sqrt{ (x_1\ – x_2)^2 + (y_1\ – y_2)^2 }\ \hspace{8cm}\]
\[=\ \sqrt{ (5\ +\ 1)^2\ +\ (2\ +\ 2)^2 }\ \hspace{8cm}\]
\[ =\ \sqrt{ ((6)^2 + (4)^2 )}\ \hspace{8cm}\]
\[ =\ \sqrt{ (36\ +\ 16 )}\ \hspace{8cm}\]
\[CD = \sqrt{52}\ \hspace{8cm}\]
\[DA\ =\ \sqrt{ (x_1\ – x_2)^2 + (y_1\ – y_2)^2 }\ \hspace{8cm}\]
\[=\ \sqrt{ (-\ 1\ -\ 3)^2\ +\ (-\ 2\ -\ 4)^2 }\ \hspace{8cm}\]
\[ =\ \sqrt{ ((-\ 4)^2 + (-\ 6)^2 )}\ \hspace{8cm}\]
\[ =\ \sqrt{ (16\ +\ 36 )}\ \hspace{8cm}\]
\[DA\ = \sqrt{52}\ \hspace{8cm}\]
\[∴\ AB = BC = CD = DA\]
\[AC\ =\ \sqrt{ (x_1- x_2)^2 + (y_1- y_2)^2 }\ \hspace{8cm}\]
\[=\ \sqrt{ (3\ -\ 5)^2\ +\ (4\ -\ 2)^2 }\ \hspace{8cm}\]
\[ =\ \sqrt{ ((-\ 2)^2 + (2)^2 )}\ \hspace{8cm}\]
\[ =\ \sqrt{ (4\ +\ 4 )}\ \hspace{8cm}\]
\[AC\ = \sqrt{8}\ \hspace{8cm}\]
\[BD\ =\ \sqrt{ (x_1- x_2)^2 + (y_1- y_2)^2 }\ \hspace{8cm}\]
\[=\ \sqrt{ (9\ +\ 1)^2\ +\ (8\ +\ 2)^2 }\ \hspace{8cm}\]
\[ =\ \sqrt{ ((10)^2 + (10)^2 )}\ \hspace{8cm}\]
\[ =\ \sqrt{ (100\ +\ 100 )}\ \hspace{8cm}\]
\[BD\ = \sqrt{200}\ \hspace{8cm}\]
\[∴\ AC \neq BD\]
\[∴\ The\ given\ complex\ numbers\ form\ a\ rhombus\]