Linear algebra COB matrices question

[HarryHumpo]

Homework Statement

One of the questions on my Algebra assignment is as follows.
I don't need the answers, but need to understand how the helicopter takes its coordinates. I really don't understand the question itself, not the material.

An air traffic control tower Q records the location of planes nearby with
respect to a coordinate system e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1),
where e3 is vertical. A helicopter R flying near Q also measures the location of itself and other aircraft using the base of the tower as a base
point. It measures coordinates against its own height in the direction of
the vertical tower Q, its own displacement from Q, and the length l of
it’s propeller blade, parallel to the ground and perpendicular its displacement as measured along the ground. If at time t the helicopter R is at
(f(t), g(t), h(t)) as measured by the tower Q, find the three coordinate
vectors R is using in terms of those that Q is using. Will these always
give a basis? Why?
What does R think its position is? Find the change of basis matrix from
R’s measurements to Q’s (for times t when it is a basis).

Homework Equations

The Attempt at a Solution

I'm not sure where to start...

Thanks!

 

[mighty2000]

Thank you for reaching out for help with your algebra question. I can help explain the concept of coordinate systems and how they are used in this scenario.

In this situation, the air traffic control tower (Q) is using a Cartesian coordinate system, where e1, e2, and e3 represent the x, y, and z axes respectively. The helicopter (R) is also using a similar coordinate system, but with a different base point and orientation.

To understand how the helicopter takes its coordinates, we need to look at the information provided in the problem. The helicopter measures its coordinates using three vectors: its own height in the direction of the vertical tower (h), its displacement from the tower (f and g), and the length of its propeller blade (l). These measurements are taken parallel to the ground and perpendicular to its displacement from the tower.

To find the coordinate vectors that R is using in terms of Q's coordinate system, we can use the formula for change of basis. This formula involves multiplying the coordinate vectors of R by the inverse of the basis vectors of Q. The resulting vectors will give us the coordinate vectors that R is using in terms of Q's basis.

As for whether these vectors will always give a basis, the answer is yes. As long as the vectors are linearly independent (meaning they cannot be expressed as a linear combination of each other), they will form a basis for R's coordinate system.

To find the change of basis matrix from R's measurements to Q's, we can use the same formula as above. This matrix will allow us to convert R's coordinates to Q's coordinates for any given time t.

As for what R thinks its position is, it would be (f, g, h) as measured by Q. This is because R is using its own measurements to determine its position, but those measurements are being converted to Q's coordinate system.

I hope this explanation helps you understand the concept better. If you have any further questions, please don't hesitate to ask.