This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back Mimi — been a while! Let's pick up where you left off. Given the complex numbers: Z_1 = 3 + 2i Z_2 = 2 - i We will find the required expressions. a) Find Z_1 + Z_2. Step 1: Substitute the given values of Z_1 and Z_2. Z_1 + Z_2 = (3 + 2i) + (2 - i) Step 2: Combine the real parts and the imaginary parts. Z_1 + Z_2 = (3 + 2) + (2i - i) Z_1 + Z_2 = 5 + i The sum is 5 + i b) Find Z_2 - Z_1. Step 1: Substitute the given values of Z_1 and Z_2. Z_2 - Z_1 = (2 - i) - (3 + 2i) Step 2: Distribute the negative sign and combine the real parts and the imaginary parts. Z_2 - Z_1 = 2 - i - 3 - 2i Z_2 - Z_1 = (2 - 3) + (-i - 2i) Z_2 - Z_1 = -1 - 3i The difference is -1 - 3i c) Find 2Z_1 + 3Z_2. Step 1: Multiply Z_1 by 2 and Z_2 by 3. 2Z_1 = 2(3 + 2i) = 6 + 4i 3Z_2 = 3(2 - i) = 6 - 3i Step 2: Add the results. 2Z_1 + 3Z_2 = (6 + 4i) + (6 - 3i) Step 3: Combine the real parts and the imaginary parts. 2Z_1 + 3Z_2 = (6 + 6) + (4i - 3i) 2Z_1 + 3Z_2 = 12 + i The result is 12 + i d) Find Z_1 Z_2. Step 1: Substitute the given values of Z_1 and Z_2. Z_1 Z_2 = (3 + 2i)(2 - i) Step 2: Multiply the complex numbers using the distributive property (FOIL). Z_1 Z_2 = 3(2) + 3(-i) + 2i(2) + 2i(-i) Z_1 Z_2 = 6 - 3i + 4i - 2i^2 Step 3: Substitute i^2 = -1 and simplify. Z_1 Z_2 = 6 + i - 2(-1) Z_1 Z_2 = 6 + i + 2 Z_1 Z_2 = 8 + i The product is 8 + i e) Find (Z_1)/(Z_2). Step 1: Substitute the given values of Z_1 and Z_2. (Z_1)/(Z_2) = (3 + 2i)/(2 - i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2 - i is 2 + i. (Z_1)/(Z_2) = ((3 + 2i)(2 + i))/((2 - i)(2 + i)) Step 3: Multiply the terms in the numerator. (3 + 2i)(2 + i) = 3(2) + 3(i) + 2i(2) + 2i(i) = 6 + 3i + 4i + 2i^2 = 6 + 7i + 2(-1) = 6 + 7i - 2 = 4 + 7i Step 4: Multiply the terms in the denominator. (2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 Step 5: Combine the numerator and denominator and express in the form a + bi. (Z_1)/(Z_2) = (4 + 7i)/(5) (Z_1)/(Z_2) = (4)/(5) + (7)/(5)i The quotient is (4)/(5) + (7)/(5)i f) Find |Z_1|. Step 1: Recall the formula for the modulus of a complex number z = a + bi, which is |z| = sqrt(a^2 + b^2). For Z_1 = 3 + 2i, we have a = 3 and b = 2. Step 2: Substitute the values into the formula. |Z_1| = sqrt(3^2 + 2^2) |Z_1| = sqrt(9 + 4) |Z_1| = sqrt(13) The modulus is sqrt(13) Got more? Send 'em!