The image shows a 3D geometry problem asking which pairs of vectors are skew lines. Skew lines are lines that are neither parallel nor intersecting. Let's analyze the options given:

1. Vectors a and b: These vectors lie in the same plane (the shaded plane on the right). Since they lie in the same plane, they cannot be skew. Therefore, a and b are not skew lines.

2. Vectors a and d: Vector $a$ lies in the plane on the right, while vector $d$ lies in the plane on the left (the larger shaded plane). Since these two vectors do not lie in the same plane and do not intersect, they are skew lines. Therefore, a and d are skew lines.

3. Vectors b and c: These vectors lie in the same plane on the right. Since they lie in the same plane, they cannot be skew. Therefore, b and c are not skew lines.

4. Vectors e and c: Vector $e$ lies in the larger shaded plane on the left, while vector $c$ lies in the right plane. Since these two vectors do not lie in the same plane and do not intersect, they are skew lines. Therefore, e and c are skew lines.

5. Vectors e and d: These vectors both lie in the same plane (the larger shaded plane on the left). Since they lie in the same plane, they cannot be skew. Therefore, e and d are not skew lines.

6. Vectors a and e: Vector $a$ lies in the plane on the right, while vector $e$ lies in the plane on the left. Since these two vectors do not lie in the same plane and do not intersect, they are skew lines. Therefore, a and e are skew lines.

Correct Answers:

• a and d
• e and c
• a and e

These pairs of vectors are skew lines.

Do you have any questions or need further details?

Related Questions:

1. What is the difference between parallel, intersecting, and skew lines?
2. Can skew lines exist in a 2D plane?
3. How do we determine if two vectors are parallel?
4. What is the significance of a plane in vector geometry?
5. How can the dot product of vectors help in identifying perpendicularity?

Tip:

When dealing with 3D geometry, always check if two lines lie in the same plane before concluding whether they are parallel, intersecting, or skew.