Here's an application-based question based on the given vectors:

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Question:

An aircraft needs to navigate in 3D space. It is currently flying along the vector $(1, 0, 0)$, representing due east. Two radio beacons are set up at nearby stations. Each of these beacons transmits signals along vectors of unit length. These vectors make an angle of 45 degrees with the direction of the aircraft's current flight path and are perpendicular to the vertical axis, represented by the vector $(0, 0, 1)$.

The aircraft's onboard computer receives the signal from both beacons and needs to determine the direction of both beacons.

1. Write down the mathematical form of these two vectors.
2. How can the aircraft use these two vectors to adjust its flight path to fly in the direction exactly halfway between the two beacons?
3. What is the angle between the new flight path of the aircraft (halfway direction) and its original direction due east?
4. Explain how the perpendicularity to $(0, 0, 1)$ helps in determining the vectors' coordinates.
5. How might atmospheric conditions, like wind, impact the flight path adjustment using these vectors?

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Do you want any details or have any questions about this problem setup?

Here are 5 related questions you could explore:

1. How do you compute the cross product of two vectors in 3D space?
2. What is the significance of a vector's magnitude and direction in navigation?
3. How does the angle between vectors affect their dot product?
4. What methods are used to find a unit vector from any arbitrary vector?
5. How do external forces like wind impact the adjustment of a flight path in navigation systems?

Tip: When solving 3D vector problems, visualizing the vectors in space can help clarify their relationships and angles.