Question

1.14. Точки A, B, C, D не лежать в одній площині. Доведіть, що прямі AC i BD не перетинаються.

Answer 1

Розв'язання завдання додаю

Answer 2

Proof that lines AC and BD do not intersect when points A, B, C, and D are not in the same plane

To prove that lines AC and BD do not intersect when points A, B, C, and D are not in the same plane, we can use a proof by contradiction.

Assume that lines AC and BD do intersect. This means that there exists a point E that lies on both lines AC and BD.

Since point E lies on line AC, we can express it as a linear combination of points A and C: E = λA + (1-λ)C, where λ is a scalar.

Similarly, since point E lies on line BD, we can express it as a linear combination of points B and D: E = μB + (1-μ)D, where μ is a scalar.

Now, let's equate the two expressions for E: λA + (1-λ)C = μB + (1-μ)D

Rearranging the equation, we get: λA - μB = (μ-λ)D - (1-λ)C

This equation represents a linear combination of vectors A and B on the left side, and a linear combination of vectors D and C on the right side.

If points A, B, C, and D are not in the same plane, then the vectors A and B are linearly independent of the vectors C and D. This means that the only way for the equation to hold true is if the coefficients λ and μ are both zero.

However, if λ = μ = 0, then point E is equal to both A and C, as well as both B and D. This implies that points A, B, C, and D are all in the same plane, which contradicts the given condition that they are not.

Therefore, our assumption that lines AC and BD intersect is false. Hence, we can conclude that lines AC and BD do not intersect when points A, B, C, and D are not in the same plane.