Resolving Vector Along Non-Standard Axes

In summary, The conversation is about resolving a vector along two axes, a and b, that are not at a right angle. The method used is to create a triangle and use the law of sines to find the projections of the vector onto each axis. The results are logically and graphically consistent and the textbook examples are limited, so verification is requested. The solution is correct but can also be shown as a linear combination of two unit vectors, \vec{a} and \vec{b}.
  • #1
srg
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Hi guys,

I have a problem in which I have to resolve [itex]\vec{R}[/itex] along two axes, a and b. However, those axes don't have a right angle between them (hence, non-standard). See the image below.

http://srg.sdf.org/images/PF/VectorHW.png [Broken]

I believe I'm doing this correctly, however my textbook has very limited examples and I'd like to verify my work.

My method for solving this is to create a triangle by "moving" the axes around and then solving for the two components of the vector.

http://srg.sdf.org/images/PF/VectorHW2.png [Broken]

In which case, using the law of sines, I get:
[tex]\frac{R_a}{\sin{110}} = \frac{800}{\sin{40}} \therefore R_a=1169.5[/tex]
[tex]\frac{R_b}{\sin{30}} = \frac{800}{\sin{40}} \therefore R_b=622.3[/tex]

Thinking about the results logically/graphically, it seems to make sense that [itex]R_a[/itex] has a higher magnitude than [itex]R_b[/itex] and that the two components make up the proper angle for [itex]\vec{R}[/itex].

Again, I believe this to be correct, however my textbook as limited examples and I'd like to confirm.

Thank you PF!
 
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  • #2
As a check... R has no vertical component so do the vertical components of Ra and Rb sum to zero?
 
  • #3
The fact that there are two lines is not really important at first- just find the projection of the given vector on each separately. To find the projection of the vector onto a imagine dropping a perpendicular from the tip of the given vector to a. That gives a right triangle with angle 30 degrees and hypotenuse of length 800N. Its projection onto a is the "near side", 800 cos(30). Similarly, the projection of the give vector on b is 800 cos(-110)= 800 cos(110)= -800 cos(20). [itex]\vec{R}= (800 cos(30))\vec{a}- (800 cos(20))\vec{b}[/itex] where [itex]\vec{a}[/itex] and [itex]\vec{b}[/itex] are unit vectors in the directions of lines a and b, respectively.
 
  • #4
resvec.JPG
You solved the problem correctly, but I would show a different picture. Resolving the vector R means that you write it as the linear combination [itex]R= R_a \hat a + R_b \hat b [/itex], sum of two vectors, parallel to a and b like in the attached picture.
 
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  • #5


Hi there,

Your method for solving this problem is correct. By using the law of sines, you have correctly determined the components of \vec{R} along the non-standard axes a and b. Your graphical representation also helps to visualize the problem and confirm your solution. Well done!

It's always good to double check your work and seek confirmation when needed. Keep up the good work in your studies and don't hesitate to ask for help when needed. Good luck with your future scientific endeavors!
 

1. What is "Resolving Vector Along Non-Standard Axes"?

"Resolving Vector Along Non-Standard Axes" is a mathematical process used to split a vector into its component parts along different axes, rather than the traditional x, y, and z axes.

2. Why is it necessary to resolve vectors along non-standard axes?

Resolving vectors along non-standard axes allows for a more accurate representation of the direction and magnitude of a vector in a specific context or coordinate system. This can be especially useful in fields such as engineering and physics where vectors may need to be analyzed in relation to specific forces or movements.

3. How is vector resolution along non-standard axes calculated?

To resolve a vector along non-standard axes, the vector is broken down into its component parts along the desired axes using trigonometric functions (such as sine and cosine) and the Pythagorean theorem.

4. Can a vector be resolved along more than one set of non-standard axes?

Yes, a vector can be resolved along multiple sets of non-standard axes. This can be useful in cases where a vector is acting in different directions simultaneously.

5. Are there any limitations to resolving vectors along non-standard axes?

One limitation is that the resolution of a vector along non-standard axes is dependent on the chosen axes and may differ from the resolution along the traditional x, y, and z axes. Additionally, the process can become more complex with multiple sets of non-standard axes or when dealing with three-dimensional vectors.

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