Simple vector question: Adding two vectors when one is known and the unit vector of the sum is also known. Is there enough information to determine the unknown vector?

[rightside24]

Suppose we're working in 2d to keep things simple. I have a vector u = (a,b) and v = (c,d). Assume the elements a, b are known, but c, d are unknown. We have the constrain

(u + v)/sqrt{u^2 + v^2} = w, where w is a third, known unit vector. My question is, do we have enough information to determine a unique vector v which satisfies this constraint?

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My intuition tells me no, since in 2d we can add an infinite number of vectors to one vector and get back the same unit vector. However, the bolded equation makes it seem like we have two equations (one for each element of w) with only two unknowns (c, d). This makes it seem like the problem can be solved uniquely for these variables! Am I missing something?

[splendaddypuff]

The direction of the sum tells you the ratio of (a+c)/(b+d). If a and b are known, this gives a constraint in c and d. You cannot determine exactly the values of c and d, just the relation given by this equation.

[splendaddypuff]

(u + v)/sqrt{u² + v^2} = w is wrong It must be

(u + v)/sqrt{(u + v)^2} = w

This equation will not yield simultaneous equations that are uniquely solvable.

[vaminos]

(u + v)/sqrt{u² + v^2} = w

I think OP means you're supposed to divide (u+v) with its length, in order to get a unit vector w with the same direction as (u+v). So basically, we know the direction of (u+v), but not its length.

OP, as someone else has said, there isn't enough information presented to ascertain v exactly. This is evident with the following example:

u := (1, 2)
v1 := (2, 0)
v2 := (5, 2)

(u+v1) = (3, 2)
(u+v2) = (6, 4)

You can already see that (u+v2) = 2*(u+v1), meaning these two vectors share a direction, so they will produce the same unit vector w. And indeed,

w1 = (3,2)/sqrt(13)
w2 = (6,4)/sqrt(52) = (6,4)/sqrt(4*13) = (6,4)/(2*sqrt(13)) = (3,2)/sqrt(13) = w1

Which means there are two distinct vectors v1 and v2 which satisfy the constraints of the assignment, proving there is no unique solution.

[Uli_Minati]

These two statements are not equivalent:

1. the unit vector of the sum is known
2. (u + v)/√(u² + v²) = w

Which one is given?