Let's solve this task step-by-step.

Task 2.25

The problem states: The intersection of planes $3x + 2y - z + 1 = 0$ and $x + 3y - 2z + 2 = 0$ is a line.

1. Part (a): Find the parametric equation of the line.
2. Part (b): Find the implicit equation of a plane that contains the line from (a) and is perpendicular to the plane $z = 0$.

Solution

Part (a): Parametric Equation of the Line

To find the line of intersection between the two planes, we solve the system of equations: $3x + 2y - z + 1 = 0$ $x + 3y - 2z + 2 = 0$

Step 1: Eliminate $z$ by manipulating the equations.

1. Multiply the first equation by 2: $6x + 4y - 2z + 2 = 0$
2. Add this to the second equation: $(6x + 4y - 2z + 2) + (x + 3y - 2z + 2) = 0$ Simplifying, we get: $7x + 7y + 4 = 0 \Rightarrow x + y = -\frac{4}{7}$

Step 2: Solve for $x$ in terms of $y$ and $z$.

Let $t$ be a parameter. Choose $x = t$. Then, $y = -5t$ and [