Decomposition of Vector Addition to lead to unique solution

[mostarac2487]

I have a puzzling question that a gentleman discussed and have differnt view on the problem, the probelm goes as follows:

What is know is the Resultant of a Vector Addition, the goal is to find the two vectors from which the resultant came from. The other piece of information that is know is that one of the vectors is held in place and the other vector is rotated 90, 180, 275 degress and the resultant remainds constant.

My thought of this is that there is more then one answer to the solution and there is no one unique solutions. Beacuse different combinations exists for one single resultant and the rotation of these angles does not play a part.

I hope this makes sense, will clarify if needed!

Thank for your input.

 

[chiro]

Hey mostarac2487 and welcome to the forums.

So we are given a + b = c where we know and want to find a and b. We also know that if we fix one vector a, and rotate b either multiples of 90 degrees without changing length that c doesn't change.

Now for this problem the thing is that for a rotation, you need to specify an axis or a plane that the rotation is happening in, which you have not. In some rotations, you can set up the rotation in a way that if you try and rotate it by any angle nothing changes and this happens when the axis of rotation and the vector you are rotating are linearly dependent (i.e. parrallel).

So let's stick to a 2D example for the moment. Let your a vector be (ax,ay) your b vector (bx,by) and your c vector (cx,cy).

Now cx = ax + bx and cy = ay + by.

Lets find out the values of rotating b for 90, 180, and 270 degrees by multiplying by the square root of -1. This gives us:

b_90 = i * (bx,by) = (-by,bx)
b_180 = i* (-by,bx) = (-bx,-by)
b_270 = i* (-bx,-by) = (by,-bx)

Now we know that for all these b's the a + b = c must hold. So let's look at the consequences of this:

ax + bx = cx, ay + by = cy implies for all b's that

ax + bx = cx, ay + by = cy (0 degrees rotation)
ax - by = cx, ay + bx = cy (90 degrees rotation)
ax - bx = cx, ay - by = cy (180 degrees rotation)
ax + by = cx, ay - bx = cy (270 degrees rotation).

This means bx = by, 2bx = 0, 2by = 0 and so on. This means that bx and by must equal 0 which means we only have one solution with two-dimensional co-ordinates.

The easiest way to think about this is that if take the zero vector and rotate it, you will always have the zero vector which has no length to begin with so you are effectively "rotating a point" which doesn't change a thing.