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The polar form of the complex numbers are z₁ = 6 · and z₂ = √2 ·
. The multiplication of the numbers is z₁ · z₂ = 6√2 ·
, the division of the numbers is z₁ / z₂ = 3√2 ·
and the reciprocal of a number is 1 / z₁ = (1 / 6) ·
.
<h3>How to find complex numbers in polar forms and make operations with them </h3>
In this problem we find two complex numbers in rectangular form (z = a + i b), whose polar form has to be found. The complex form is described below:
z = r ·
Where:
- r - Norm
- θ - Direction, in radians.
The magnitude is determined by Pythagorean theorem:
r = √(a² + b²)
And the direction by inverse trigonometric functions:
θ = tan⁻¹ (b / a)
Polar form offers a quicker manner to perform multiplication and division of complex numbers. Thus:
Multiplication
z₁ · z₂ = r₁ · r₂ ·
Division
z₁ / z₂ = (r₁ / r₂) ·
First, determine the magnitudes of each complex number:
r₁ = √[(3√3)² + (- 3)²]
r₁ = 6
r₂ = √[(- 1)² + 1²]
r₂ = √2
Second, determine the directions of each complex number:
θ₁ = 11π / 6
θ₂ = 3π / 4
Third, write the complex numbers in polar form:
z₁ = 6 ·
z₂ = √2 ·
Fourth, find the multiplication of the complex numbers:
z₁ · z₂ = 6√2 ·
Fifth, find the division of the complex numbers:
z₁ / z₂ = 3√2 ·
Sixth, find the reciprocal of the complex number:
1 / z₁ = (1 / 6) ·
To learn more on complex numbers: brainly.com/question/20566728
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