Question

given ( z_1 = 9 + 9sqrt{3}i ) and ( z_2 = 2 + sqrt{3}i ), what is ( z_2 - z_1 )?
( 8 + 7sqrt{3}i )
( -7 - 8sqrt{3}i )
( 7 + 10sqrt{3}i )
( -7 + 10sqrt{3}i )

Explanation:

Step1: Substitute the values of \( z_1 \) and \( z_2 \) into \( z_2 - z_1 \)

We have \( z_1 = 9 + 9\sqrt{3}i \) and \( z_2 = 2 + \sqrt{3}i \). So, \( z_2 - z_1=(2 + \sqrt{3}i)-(9 + 9\sqrt{3}i) \).

Step2: Distribute the negative sign

Using the distributive property \( a-(b + c)=a - b - c \), we get \( 2 + \sqrt{3}i-9 - 9\sqrt{3}i \).

Step3: Combine the real parts and the imaginary parts separately

For the real parts: \( 2-9=-7 \). For the imaginary parts: \( \sqrt{3}i-9\sqrt{3}i=(1 - 9)\sqrt{3}i=-8\sqrt{3}i \).

Step4: Combine the results

Combining the real and imaginary parts, we have \( z_2 - z_1=-7-8\sqrt{3}i \).

Answer:

\( -7 - 8\sqrt{3}i \) (corresponding to the option with this value)