We are tasked with finding the parametric form of the vector equation and Cartesian equations for a line passing through the given points and parallel to a given line.

Given Information:

1. Line equation: $\frac{x-2}{2} = \frac{y-2}{3} = \frac{z-1}{3}$
2. The new line is parallel to the line: $\frac{x-1}{3} = \frac{-y+1}{2} = \frac{z+1}{1}$

Step 1: Extracting Direction Ratios from the Given Line

From the line $\frac{x-1}{3} = \frac{-y+1}{2} = \frac{z+1}{1}$, we can extract the direction ratios as: $\langle 3, -2, 1 \rangle$ These are the components of the direction vector of the line.

Step 2: Parametric Form of the Given Line

We rewrite the original line $\frac{x-2}{2} = \frac{y-2}{3} = \frac{z-1}{3}$ in parametric form. Let $t$ be a parameter: $x = 2 + 2t, \quad y = 2 + 3t, \quad z = 1 + 3t$ This is the parametric form of the given line.

Step 3: Parametric Form of the New Line

The new line passes through the point $(2, 2, 1)$, and since it's parallel to the given line, it shares the same direction ratios. Therefore, the parametric form of the new line is: $x = 2 + 3t, \quad y = 2 - 2t, \quad z = 1 + t$ where $t$ is the parameter.

Step 4: Cartesian Form of the New Line

To convert this parametric form back into Cartesian form, we express $t$ in terms of $x$, $y$, and $z$:

• From $x = 2 + 3t$, we get: $t = \frac{x - 2}{3}$
• From $y = 2 - 2t$, we get: $t = \frac{2 - y}{2}$
• From $z = 1 + t$, we get: $t = z - 1$

Equating these expressions for $t$, we get the Cartesian equations: $\frac{x - 2}{3} = \frac{2 - y}{2} = z - 1$

Final Answer:

• Parametric Form: $x = 2 + 3t$, $y = 2 - 2t$, $z = 1 + t$
• Cartesian Form: $\frac{x - 2}{3} = \frac{2 - y}{2} = z - 1$

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Would you like further details or explanations on any steps?

Here are 5 additional questions for you to consider:

1. How would you find the angle between two lines given their direction vectors?
2. Can you derive the direction ratios from parametric equations?
3. How would you confirm if two lines are parallel or skew?
4. What is the vector equation of a line passing through two points?
5. How can you find the shortest distance between two skew lines?

Tip: The direction ratios of a line can be found directly from the parametric form of the line by analyzing the coefficients of the parameter.