Find parametric equations for the line through the point 
(0, 3, 2)
 that is perpendicular to the line 
x = 1 + t, y = 3 − t, z = 3t
 and intersects this line

To find the parametric equations of a line that passes through the point \( (0, 3, 2) \), is perpendicular to the line \( x = 1 + t, y = 3 - t, z = 3t \), and intersects that line, we need to follow these steps:

### Step 1: Parametric Equation of the Given Line
The parametric equations of the given line are:

\[
x = 1 + t, \quad y = 3 - t, \quad z = 3t
\]

This means that any point \( P(t) \) on this line can be expressed as:
\[
P(t) = (1 + t, 3 - t, 3t)
\]

### Step 2: Vector Form of the Given Line
The direction vector of the given line is derived from the coefficients of \( t \) in the parametric equations:

\[
\mathbf{v_1} = (1, -1, 3)
\]

This is the direction of the given line.

### Step 3: Direction of the New Line
Let \( \mathbf{v_2} = (a, b, c) \) be the direction vector of the line we are looking for. Since the two lines are perpendicular, their direction vectors must satisfy the condition that their dot product is zero:

\[
\mathbf{v_1} \cdot \mathbf{v_2} = 0
\]

That is:
\[
1 \cdot a + (-1) \cdot b + 3 \cdot c = 0
\]

Simplifying:
\[
a - b + 3c = 0 \tag{1}
\]

### Step 4: Intersection Condition
Let \( P(t_0) = (1 + t_0, 3 - t_0, 3t_0) \) be the point where the two lines intersect. Since the new line passes through \( (0, 3, 2) \) and intersects the given line, this point must satisfy the parametric equations of the new line. 

Let the parametric equations of the new line be:
\[
x = 0 + a s, \quad y = 3 + b s, \quad z = 2 + c s
\]

At the point of intersection, these coordinates should be equal to \( P(t_0) \):

\[
0 + a s = 1 + t_0 \tag{2}
\]
\[
3 + b s = 3 - t_0 \tag{3}
\]
\[
2 + c s = 3t_0 \tag{4}
\]

### Step 5: Solving the System
Now, you have a system of equations:

- \( a s = 1 + t_0 \)
- \( b s = -t_0 \)
- \( c s = 3t_0 - 2 \)

Simultaneously solve this system to find \( s \), \( t_0 \), and the direction vector \( (a, b, c) \).

---

Would you like further clarification or a step-by-step breakdown of the algebra involved in solving this system?

### 5 Related Problems:
1. Find the parametric equations of a line perpendicular to two given lines.
2. Determine the intersection point of two skew lines.
3. How do you determine if two lines are parallel, intersecting, or skew?
4. Given a line and a point, find the distance from the point to the line.
5. Derive the parametric form of a plane equation given three points.

**Tip:** When dealing with perpendicular vectors, always use the dot product to impose orthogonality conditions.

Find parametric equations for the line through the point (0, 3, 2) that is perpendicular to the line x = 1 + t, y = 3 − t, z = 3t and intersects this line.

Learn how to find the parametric equations of a line through the point (0, 3, 2) that is perpendicular to the line given by x = 1 + t, y = 3 − t, z = 3t. This problem involves parametric equations, the dot product for perpendicularity, and finding the intersection of lines in 3D geometry. Suitable for high school students in Grades 11-12.

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