To solve this problem, let's analyze each answer choice and determine if it represents the intersection of three planes, which should be either a line or a single point if the planes intersect uniquely.

Solution Analysis:

1. Intersection of Three Planes: When three planes intersect, their intersection can be:

   • A single point, if they intersect uniquely.
   • A line, if two planes intersect along a line, and the third plane intersects this line.

2. Evaluating the Choices: Let's go through each option to see if it represents a line or a point.

   • Option (a):
     $(x, y, z) = (1, 1, 0) + t(2, 2, 0) + s(-2, 4, 0), \quad s, t \in \mathbb{R}$ This represents a parametric form of a plane rather than a line or a point because it has two free parameters ($s$ and $t$) that span a two-dimensional surface. This does not represent a line or a point, so it cannot be the intersection of three planes.

   • Option (b): $(x, y, z) = (2, 1, 0) + t(1, 2, 0) + s(-2, -4, 0), \quad s, t \in \mathbb{R}$ This also represents a parametric form of a plane because it has two parameters ($s$ and $t$), which span a plane rather than a line or point. So, this does not represent the intersection of three planes either.

   • Option (c): $x = 1, \quad y = 2, \quad z = 3$ This set of equations represents a single point at $(1, 2, 3)$. If three planes intersect at exactly one point, this would be the result. Therefore, this choice is a possible solution, as it aligns with a unique intersection point of three planes.

   • Option (d): $x + 2y - 3z = 5$ This is the equation of a single plane, not a line or a point. A single plane alone cannot represent the intersection of three planes, so this is also not a valid answer.

Conclusion:

The correct answer is (c), as it is the only choice that represents a unique intersection point, which matches the possible result of three planes intersecting at a single point.

Would you like more details on the intersection of planes or help with a similar problem?

Follow-up Questions:

1. What conditions determine if three planes intersect at a single point versus a line?
2. How can we convert parametric equations of a plane into standard form?
3. What does it mean for planes to be parallel, and how does this affect their intersections?
4. How would the intersection change if two planes are identical?
5. How can we verify if three planes are coplanar?

Tip:

For intersections involving parametric equations, look at the number of free parameters. If there's only one, it’s a line; if there are two, it’s a plane.