To simplify the expression $\frac{6 - 7i}{3i}$, we follow a systematic process:

Step 1: Multiply both the numerator and the denominator by the complex conjugate of the denominator, which in this case is $i$ (since the denominator is $3i$).

$\frac{6 - 7i}{3i} \times \frac{i}{i} = \frac{(6 - 7i) \cdot i}{(3i) \cdot i}$

Step 2: Simplify the denominator:

$(3i) \cdot i = 3i^2 = 3(-1) = -3$

Step 3: Distribute $i$ in the numerator:

$(6 - 7i) \cdot i = 6i - 7i^2$ Since $i^2 = -1$, we have:

$6i - 7(-1) = 6i + 7$

Step 4: Rewrite the fraction:

$\frac{6i + 7}{-3}$

Step 5: Split the fraction:

$\frac{6i}{-3} + \frac{7}{-3} = -2i - \frac{7}{3}$

Thus, the simplified result is:

$-\frac{7}{3} - 2i$

Final Answer:

$\boxed{-\frac{7}{3} - 2i}$

Would you like further details or have any questions on this process?

Here are 5 related questions you might find interesting:

1. What is the conjugate of a complex number and why is it useful in simplifying expressions?
2. How do we handle division of complex numbers involving real and imaginary parts?
3. What does the notation $i$ represent in complex numbers?
4. Can you explain the process of multiplying complex numbers and distributing terms?
5. How do you simplify expressions involving $i^2$?

Tip: When dividing complex numbers, always multiply both the numerator and denominator by the conjugate of the denominator to eliminate complex numbers in the denominator.