Vectors - Magnitude of two vectors relative to each other

[Jackist]

I know there is a way to work this out using a vector diagram (which I don't fully understand still) but I was wondering if there is a purely mathematical way to solve this also? Instructions on the process to solve this one way or both would be great please. I am not just looking for the answers.

The best way to visualise this is to think of A and B as two towns far apart and C as just a fixed point in the same country which A and B are moving away from as a result of tectonics.

This is a two part Question:

A) Two GPS sites, A and B, are moving with respect to a common reference site, C, with the following velocities: vA-C = 20 mm/yr at an azimuth of 140 deg; vB-C = 30 mm/yr at an azimuth of 225 deg. By constructing a vector diagram representing these velocities (or otherwise), calculate the magnitude (in mm/yr) of the velocity of point A relative to point B, vA-B.

B) Using the information given in the previous question, calculate the direction (in degrees) of the velocity of A with respect to B, vA-B.

Thanks heaps

 

[srmichael]

I know there is a way to work this out using a vector diagram (which I don't fully understand still) but I was wondering if there is a purely mathematical way to solve this also? Instructions on the process to solve this one way or both would be great please. I am not just looking for the answers.

The best way to visualise this is to think of A and B as two towns far apart and C as just a fixed point in the same country which A and B are moving away from as a result of tectonics.

This is a two part Question:

A) Two GPS sites, A and B, are moving with respect to a common reference site, C, with the following velocities: vA-C = 20 mm/yr at an azimuth of 140 deg; vB-C = 30 mm/yr at an azimuth of 225 deg. By constructing a vector diagram representing these velocities (or otherwise), calculate the magnitude (in mm/yr) of the velocity of point A relative to point B, vA-B.

B) Using the information given in the previous question, calculate the direction (in degrees) of the velocity of A with respect to B, vA-B.

Thanks heaps

Some teacher's try and teach this topic using a variety of triangles and introduce Law of Cosines and other confusing stuff for most kids. I think it is much easier to just calculate the component foem of each vector, then add them up and that is the component form of your resulting vector. From there you can calculate the magnitude and direction of the resultant vector.

A vector with magnitude M and direction Θ will have a component form of: (as Denis would say, tatoo this on your wrist): \(\displaystyle <M\cdot\cos\theta,M\cdot\sin\theta>\)

One thing to keep in mind, the directional angle Θ is based on standard degree measurment using the positive x-axis as 0° and then going in a counter clockwise direction. This problem, however, uses an azimuth angle, which uses the positive y-axis as North or 0° and then goes in a clockwise motion. You must convert this angel to the traditional directional angle. One can use this handy formula: Let azimuth angle = ß, then Θ = 450° - ß

I'll start off with the first vector, \(\displaystyle F_{A}\):

Θ = 450° - 140° = 310°
M = 20
\(\displaystyle F_{A}=<20\cdot\cos(310°),20\cdot\sin(310°)>=<12.86,-15.32>\)

Do the same for vector B, add them up, then you can find the magnitude and direction. Don't forget to change your directional angle Θ to an azimuth angle ß by rearranging the formula above and using ß = 450° - Θ.

Let us know what you get!