Question

finfd the modulus and amplitude of the following complex number
7 - 5i​

Answer 1

Answer:

$: \: \:$

Step-by-step explanation:

If z = a + ib is any complex number such that a ∈R, b∈R  

and i = √-1,

then the modulus of complex number, |Z| is given by

|Z| = √a² + b²

Amplitude of the complex number , θ is given by,

θ = tan⁻¹(b/a)

i) Z = 7 - 5i

Modulus,|Z| = √7² + 5² = √74

Amplitude,θ = tan⁻¹(-5/7) = -tan⁻¹(5/7)

Answer 2

Given:

Complex Number➺ 7-5i

To Find:

Modulus and amplitude of the given complex number

Solution:

We know that,

For a complex no., a+ib where a,b∈R

✏ $\bf{Modulus=\sqrt{a^{2}+b^{2}}}$

✏ $\bf{Argument(\theta)=tan^{-1}\bigg|\dfrac{b}{a}\bigg|}$

Here,

a= 7

b= -5

Now,

Modulus of 7-5i

= $\sqrt{7^{2}+(-5)^{2}}$

= $\sqrt{49+25}$

= $\sqrt{74}$

Now,

Amplitude,θ= $tan^{-1}\bigg|\dfrac{-5}{7}\bigg|$

θ= $tan^{-1}\dfrac{5}{7}$

Hence, Modulus and amplitude of given complex number are √74 and tan⁻¹(5/7) respectively.

Some Properties of Modulus

If z,z₁,z₂∈C, where C is set of complex numbers

↬ |z|= |-z|

↬ |z₁z₂|=|z₁||z₂|

↬ |z₁+z₂|²+|z₁-z₂|²= 2(|z₁|²+|z₂|²)