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 $(a, b, c)$.

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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.