Question

If $z_1=2+3i$ and $z_2=3+2i$, then $|z_1+z_2|$ is equal to

• A. 50
• B. 10
• C. $5\sqrt{2}$
• D. 25
• E. $2\sqrt{5}$

Hint:

Calculation Tip: Do NOT confuse $|z_1 + z_2|$ with $|z_1| + |z_2|$. The Triangle Inequality states that $|z_1 + z_2| \le |z_1| + |z_2|$, meaning you cannot simply distribute absolute value bars across addition. Always add first!

Answer 1

The Correct Option is C

Concept:
To find the modulus of the sum of two complex numbers, you must first add the numbers together by combining their real parts and imaginary parts separately. Once the sum is in the standard form $z = a + bi$, calculate its modulus using the distance formula: $|z| = \sqrt{a^2 + b^2}$.

Step 1: Identify the given complex numbers.
We are provided with two complex numbers: $$z_1 = 2 + 3i$$ $$z_2 = 3 + 2i$$

Step 2: Set up the addition expression.
To find $z_1 + z_2$, add the corresponding real parts together, and add the corresponding imaginary parts together: $$z_1 + z_2 = (2 + 3i) + (3 + 2i)$$

Step 3: Calculate the sum of the complex numbers.
Group the real and imaginary terms: $$z_1 + z_2 = (2 + 3) + i(3 + 2)$$ $$z_1 + z_2 = 5 + 5i$$

Step 4: Apply the modulus formula.
Now we need to find the modulus of the new complex number $5 + 5i$. Using $|a + bi| = \sqrt{a^2 + b^2}$, where $a = 5$ and $b = 5$: $$|z_1 + z_2| = \sqrt{5^2 + 5^2}$$

Step 5: Simplify the radical expression.
Evaluate the squares and add them: $$|z_1 + z_2| = \sqrt{25 + 25}$$ $$|z_1 + z_2| = \sqrt{50}$$ Factor out the largest perfect square to simplify: $$|z_1 + z_2| = \sqrt{25 \cdot 2} = 5\sqrt{2}$$ Hence the correct answer is (C) $5\sqrt{2$}.