The lines `(x-1)/1=(y-2)/2=(z-3)/3` and
`x/2=(y+2)/2=(z-3)/-2` are parallel.

Answer

To determine if the lines given by the equations \((x-1)/1=(y-2)/2=(z-3)/3\) and \(x/2=(y+2)/2=(z-3)/-2\) are parallel, we will follow these steps:

### Step 1: Identify the Direction Ratios of Each Line
The first line can be expressed in the symmetric form:
\[ \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3} \]
From this, we can extract the direction ratios of the first line (let's call it Line 1):
- Direction ratios of Line 1: \( (1, 2, 3) \)

The second line can be expressed as:
\[ \frac{x}{2} = \frac{y+2}{2} = \frac{z-3}{-2} \]
From this, we can extract the direction ratios of the second line (let's call it Line 2):
- Direction ratios of Line 2: \( (2, 2, -2) \)

### Step 2: Check the Condition for Parallel Lines
For two lines to be parallel, their direction ratios must be proportional. This means we need to check if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
where \( (a_1, b_1, c_1) \) are the direction ratios of Line 1 and \( (a_2, b_2, c_2) \) are the direction ratios of Line 2.

Substituting the values we found:
- \( a_1 = 1, b_1 = 2, c_1 = 3 \)
- \( a_2 = 2, b_2 = 2, c_2 = -2 \)

Now we calculate:
\[ \frac{1}{2}, \quad \frac{2}{2}, \quad \frac{3}{-2} \]

### Step 3: Evaluate the Ratios
Calculating the ratios gives us:
- \( \frac{1}{2} = 0.5 \)
- \( \frac{2}{2} = 1 \)
- \( \frac{3}{-2} = -1.5 \)

### Step 4: Compare the Ratios
Now we compare the calculated ratios:
\[ \frac{1}{2} \neq \frac{2}{2} \neq \frac{3}{-2} \]

Since these ratios are not equal, we conclude that the lines are not parallel.

### Final Conclusion
The lines given by the equations \((x-1)/1=(y-2)/2=(z-3)/3\) and \(x/2=(y+2)/2=(z-3)/-2\) are **not parallel**.

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