Question Video: The Modulus of the Product of Complex Numbers in Polar Form Mathematics

What is the magnitude of the product of 𝑍₁ = π‘Ÿ(cos πœƒ + 𝑖 sin πœƒ) and 𝑍₂ = 𝑠(cos πœ™ + 𝑖 sin πœ™)?

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Video Transcript

What is the magnitude of the product of 𝑍 sub one is equal to π‘Ÿ multiplied by the cos of πœƒ plus 𝑖 sin of πœƒ and 𝑍 sub two is 𝑠 multiplied by the cos of πœ™ plus 𝑖 sin πœ™?

In this question, we’re asked to determine the magnitude of the product of two given complex numbers. That’s 𝑍 sub one and 𝑍 sub two. And we can see these are given in polar form. And we recall this tells us the constant coefficient at the start of the expression is the magnitude of the complex number and the angle is the argument of the complex number. For example, 𝑍 sub two has magnitude 𝑠 and argument πœ™.

We can then use the properties of multiplying complex numbers given in polar form to find the magnitude of the product of these two numbers. We recall, to multiply two complex numbers given in polar form, we multiply the magnitude of both of these numbers and we add their arguments. 𝑍 sub one times 𝑍 sub two is π‘Ÿ times 𝑠 multiplied by the cos of πœƒ plus πœ™ plus 𝑖 sin of πœƒ plus πœ™. In other words, the magnitude of the product of two complex numbers is the product of their individual magnitudes. The magnitude of 𝑍 sub one times 𝑍 sub two is the magnitude of 𝑍 sub one multiplied by the magnitude of 𝑍 sub two. And we can use this to find the magnitude of this product. Since 𝑍 sub one is given in polar form, we can read off its magnitude. Its magnitude is π‘Ÿ. Similarly, we know 𝑍 sub two has magnitude 𝑠.

Therefore, the magnitude of 𝑍 sub one times 𝑍 sub two is equal to π‘Ÿ multiplied by 𝑠.

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