Represent on complex plane the complex numbers `w=3 + 4i and z= 6-3i` together with `w +z and w-z`. Obtain the modulus and argument of w and z.

Answer

To solve the problem step by step, we will represent the complex numbers \( w = 3 + 4i \) and \( z = 6 - 3i \) on the complex plane, calculate \( w + z \) and \( w - z \), and then find the modulus and argument of \( w \) and \( z \).

### Step 1: Calculate \( w + z \)
\[ w + z = (3 + 4i) + (6 - 3i) = 3 + 6 + (4i - 3i) = 9 + i \]

### Step 2: Calculate \( w - z \)
\[ w - z = (3 + 4i) - (6 - 3i) = 3 + 4i - 6 + 3i = (3 - 6) + (4i + 3i) = -3 + 7i \]

### Step 3: Plot the points on the complex plane
- For \( w = 3 + 4i \), plot the point (3, 4).
- For \( z = 6 - 3i \), plot the point (6, -3).
- For \( w + z = 9 + i \), plot the point (9, 1).
- For \( w - z = -3 + 7i \), plot the point (-3, 7).

### Step 4: Calculate the modulus of \( w \)
\[ |w| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Calculate the modulus of \( z \)
\[ |z| = \sqrt{(6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]

### Step 6: Calculate the argument of \( w \)
\[ \text{arg}(w) = \tan^{-1}\left(\frac{4}{3}\right) \]

### Step 7: Calculate the argument of \( z \)
\[ \text{arg}(z) = \tan^{-1}\left(\frac{-3}{6}\right) = \tan^{-1}\left(-\frac{1}{2}\right) \]

### Summary of Results
- \( w + z = 9 + i \)
- \( w - z = -3 + 7i \)
- Modulus of \( w = 5 \)
- Modulus of \( z = 3\sqrt{5} \)
- Argument of \( w = \tan^{-1}\left(\frac{4}{3}\right) \)
- Argument of \( z = \tan^{-1}\left(-\frac{1}{2}\right) \)