Question Video: Finding the Distance between Two Complex Numbers Using Modulus Notation Mathematics

Video Transcript

If 𝑧 one is equal to negative four minus nine 𝑖 and 𝑧 two is equal to three minus three 𝑖, what is the absolute value or modulus of 𝑧 two minus 𝑧 one.

We will begin this question by calculating 𝑧 two minus 𝑧 one. To do this, we will subtract the complex number negative four minus nine 𝑖 from three minus three 𝑖. We can simplify this expression by collecting the real and imaginary parts. Three minus negative four is the same as three plus four. Therefore, the real part of 𝑧 two minus 𝑧 one is seven. Negative three 𝑖 minus negative nine 𝑖 is the same as negative three 𝑖 plus nine 𝑖, which gives us an imaginary part equal to six 𝑖.

𝑧 two minus 𝑧 one is equal to seven plus six 𝑖. We recall that if the complex number 𝑧 is equal to 𝑎 plus 𝑏𝑖, then the modulus of 𝑧 is equal to the square root of 𝑎 squared plus 𝑏 squared. This means that the modulus of 𝑧 two minus 𝑧 one is the square root of seven squared plus six squared. Seven squared is equal to 49 and six squared is equal to 36. As the sum of these two numbers is 85, the modulus of 𝑧 two minus 𝑧 one is the square root of 85.