Vector A has the magnitude of 12 m and is angled 60 degrees counterclockwise from the positive direction of the x axis of an xy plane. Also vector b = 12i + 8j on that same corrdinate system. We now rotate the system counterclockwise about the orgin by 20 degrees to form a new system. On this new system what are (a) vector A and vector B both in unit-vector notation?
I couldn't figure out what I should do to solve this problem.
Change vector B to polar (magnitude at some angle). I think this will be 14.4 at 33.7 degrees counterclockwise from the x axis. Check that. Now, sketch the system, including the new axis. On the new system, you have
A'=12@(60-20) and B'=14.4@(33.7-20)
Convert those to the new i,j system ...
A'=12'cos40+ 12j'sin40 check that, and convert the B'
I will be happy to check your work.
If you are a college student, there is another way of rotating axis and vector systems, in vector form. Check your text on that.
http://www.nytimes.com/2006/10/06/us/06evangelical.html?_r=1&ref=us&oref=slogin
To solve this problem, you can follow these steps:
1. Start by converting vector A to polar form. Since the magnitude of vector A is 12 m and it is angled 60 degrees counterclockwise from the positive x-axis, you can represent it as A = 12 @ 60 degrees.
2. Convert vector B to polar form. Since vector B is given as B = 12i + 8j in the original coordinate system, you can calculate its magnitude as sqrt(12^2 + 8^2) = 14.4 and its angle counterclockwise from the positive x-axis as arctan(8/12) ≈ 33.7 degrees. Therefore, in polar form, vector B becomes B = 14.4 @ 33.7 degrees.
3. Sketch the new coordinate system, including the new axes resulting from the counterclockwise rotation of 20 degrees about the origin.
4. Use the formula for rotating vectors in a coordinate system:
A' = (A magnitude) @ (A angle - rotation angle),
B' = (B magnitude) @ (B angle - rotation angle).
In this case, the rotation angle is 20 degrees.
5. Calculate the new values for vector A and vector B by substituting the magnitudes and angles into the formulas:
A' = 12 @ (60 - 20) = 12 @ 40 degrees,
B' = 14.4 @ (33.7 - 20) = 14.4 @ 13.7 degrees.
6. Convert the new vectors A' and B' to the new coordinate system in unit-vector notation.
A' = 12'cos(40) + 12'j*sin(40),
B' = 14.4'cos(13.7) + 14.4'j*sin(13.7).
7. Simplify the expressions to obtain the unit-vector notations for vector A' and vector B'.
I hope this helps! Let me know if you have any further questions.