Question

Lines a and b are skew because they are not parallel, not in the same plane, and do not intersect. This can be determined by examining the angles formed by lines a and b, as they are not equal and do not have the same degree. Additionally, lines a and b cannot be in the same plane as planes q and r because they do not intersect. Therefore, lines a and b must be skew since they are not parallel to planes q and r. Which of the following statements is true about lines a and b?

1) Lines a and b are parallel to each other.
2) Lines a and b intersect at a point.
3) Lines a and b are in the same plane as planes q and r.
4) Lines a and b are skew to each other.

Answer 1

Lines a and b are skew lines because they are not parallel, do not lie in the same plane, and do not intersect. Vectors that are perpendicular form a 90° angle, and the vector product results in a perpendicular vector to the plane of the original vectors.

Step-by-step explanation:

If lines a and b are not parallel, not in the same plane, and do not intersect, then they can be described as skew lines. Skew lines are two lines that do not intersect and are not parallel, meaning they do not lie in the same plane. The fact that angles formed by lines a and b are not equal and lines a and b are not parallel to planes q and r further suggests that they are indeed skew to each other. When we talk about vectors, if vectors point in the same direction, they are parallel (a). If they point in opposite directions, they are still parallel but in opposite senses (c). When vectors form a 90° angle with each other, they are perpendicular (b), meaning they are orthogonal, and the same applies to planes and lines. The vector product of two vectors results in a vector that is perpendicular to the plane containing the original vectors, which also aligns with the definition of skew lines in three-dimensional space.