Question

the modulus and amplitude of z=3(1+i) is ?​

Answer 1

Answer:

Modulus of complex number = $3\sqrt{2}$ and amplitude = $\frac{\pi }{4}$

Step-by-step explanation:

Let,

We have complex number Z = a + ib

Then modulus of complex number z = $\sqrt{a^{2} + b^{2} }$

then amplitude of z = $tan^{-1} (\frac{b }{a})$

If a and b belong from 1 quadrant then amplitude will be = $tan^{-1} (\frac{b }{a})$  

If a and b belong from 2 quadrant then amplitude will be = $\pi - tan^{-1} (\frac{b }{a})$

If a and b belong from 3 quadrant then amplitude will be = $- \pi + tan^{-1} (\frac{b }{a})$

If a and b belong from 4 quadrant then amplitude will be = $- tan^{-1} (\frac{b }{a})$

In question,

Z = 3(1 + i)

modulus of z = $3\sqrt{1^{2} + 1^{2} }$

= $3\sqrt{2}$

Amplitude of z will be = $tan^{-1} (\frac{1 }{1})$

And both number belong from 1 quadrant so amplitude will be in 1 quadrant

Amplitude = $tan^{-1} {1 }$

∵$tan^{-1} {1 } = \frac{\pi }{4}$

Final answer,

Modulus of Z = $3\sqrt{2}$

Amplitude of Z = $\frac{\pi }{4}$

Answer 2

Given: Complex number is z = 3(1 + i)

To find: It's modulus and argument

Solution:

Modulus of any complex number is given by root over the sum of squares of real and imaginary part of complex number.

i.e for any complex number of the form a + bi, it's modulus is given by:

$\sf |z| = \sqrt{a^2 + b^2}$

We have,

$\sf z = 3(1 + i)\\\\\implies z = 3 + 3i\\\\\implies |z| = \sqrt{3^2 + 3^2}\\\\\implies |z| = \sqrt{9 +9}\\\\\implies |z| = \sqrt{18}\\\\\implies |z| = 3\sqrt{2}$

So the modulus of the complex number is 3√2.

$\rule{280}{1}$

Argument of any complex number of the form, a + bi is given by:

$\sf \theta = \arctan\left|\dfrac{b}{a}\right|$

We have,

a = b = 3

$\sf\implies \theta = \arctan\left| \dfrac{3}{3}\right|\\\\\implies \theta = \arctan|1|\\\\\implies \theta = \dfrac{\pi}{4}$

So the argument of the complex number is pi/4.

Option [D] is correct.