Resolving Vector Along Non-Standard Axes

srg

Gold Member
Hi guys,

I have a problem in which I have to resolve $\vec{R}$ 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:
$$\frac{R_a}{\sin{110}} = \frac{800}{\sin{40}} \therefore R_a=1169.5$$
$$\frac{R_b}{\sin{30}} = \frac{800}{\sin{40}} \therefore R_b=622.3$$

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

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

Thank you PF!

Last edited by a moderator:
• PhanthomJay
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CWatters

Homework Helper
Gold Member
As a check... R has no vertical component so do the vertical components of Ra and Rb sum to zero?

HallsofIvy

Homework Helper
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). $\vec{R}= (800 cos(30))\vec{a}- (800 cos(20))\vec{b}$ where $\vec{a}$ and $\vec{b}$ are unit vectors in the directions of lines a and b, respectively.

ehild

Homework Helper 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 $R= R_a \hat a + R_b \hat b$, sum of two vectors, parallel to a and b like in the attached picture.

• srg

"Resolving Vector Along Non-Standard Axes"

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