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In this problem, we are working with a cube and are asked to determine which of the listed lines are perpendicular to the given planes.

The cube is labeled with vertices: \( A, B, C, D \) at the bottom, and \( A_1, B_1, C_1, D_1 \) at the top.

- The planes to which the perpendicularity of certain lines is to be determined are:
  1. The plane \( BCC_1 \) for part (a)
  2. The plane \( ACC_1 \) for part (b)

### Part (a) Plane \( BCC_1 \):
This is the plane formed by points \( B, C, C_1 \). A plane perpendicular to it will be a line that intersects the plane at 90 degrees. The perpendicular line to this plane is likely to be either one of the vertical edges or a diagonal.

Looking at the vertices, the line perpendicular to this plane would be the vertical edge \( BB_1 \), as it directly stands perpendicular to the base and cuts through the plane.

### Part (b) Plane \( ACC_1 \):
This is the plane formed by points \( A, C, C_1 \). The line perpendicular to this plane would be another edge that stands vertically or directly intersects the plane.

In this case, the line perpendicular to this plane is \( AA_1 \), which, like in part (a), is a vertical line passing through the vertices and is perpendicular to the plane.

### Conclusion:
- For part (a), the perpendicular line is \( BB_1 \).
- For part (b), the perpendicular line is \( AA_1 \).

Let me know if you want further clarification or details!

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Here are some related questions for better understanding:
1. What is the general definition of a plane in 3D space?
2. How do we determine if a line is perpendicular to a plane in geometry?
3. Can two lines be perpendicular to the same plane simultaneously?
4. How would this problem change if the cube were rotated in space?
5. What are the properties of diagonal lines in a cube?

**Tip**: When analyzing perpendicularity in a cube, always check if a line intersects the plane at 90 degrees by visualizing the right angle between the line and the plane's normal vector.

Given a cube, determine which of the listed lines is perpendicular to the given planes: (a) plane BCC_1, (b) plane ACC_1.

This problem involves determining which lines in a cube are perpendicular to given planes. Specifically, it covers the perpendicularity of lines BB_1 to plane BCC_1 and AA_1 to plane ACC_1. Suitable for high school students studying geometry.

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