# Angle between two vectors (Only by having their lengths)

1. Oct 28, 2015

### Driverfury

1. The problem statement, all variables and given/known data
Ok guys, I'm studying Physics and I started, of course, by learning about vectors.

So I have this problem:

There are two vectors: A (its length is 3 m) and B (its length is 4 m). I have not other informations about them.
I want to know how I can dispose them to get a vector S (which is A + B) such as:
a) its length is 7 m.
b) its length is 1 m.
c) its length is 5 m.

Now, I know that the points a and b are obvious but I want a rigorous solution which is valid for point c and for every other length of the vector S.

2. Relevant equations
Note
: From now on when I write a I mean the length of the vector A. When I write b I mean the length of the vector B. Let's call o the agle between them.

Maybe this equation will be useful.

cos(o) = (A . B) / (a b)

3. The attempt at a solution
I tried but I cannot evaluate the scalar product A . B because it is equals to (a b cos(o)). And if I replace it in the equation I have:

cos(o) = (a b cos(o)) / (a b) = cos(o)

2. Oct 28, 2015

### Orodruin

Staff Emeritus
Can you think of a way of writing the length of the composite vector using only the inner product and the lengths of A and B?

3. Oct 28, 2015

### RUber

You have $|A||B|\cos(\theta) =A \cdot B$. I am not sure that is exactly what you want to do, since C= A+B.
In theory, let's say you make A point along the x axis. A + B would be $( |A| +B_x)\hat x + B_y \hat y$ which is adding the part of B in the x direction to A and the remaining component of B is in the y direction.
$B_x$ can also be thought of as $B\cdot \hat x$ and similarly $B_y = B\cdot \hat y = B - B_x$.
In more general terms, you don't even need to align A with the x axis, you could just make $\hat a$ a unit vector in the A direction.

If that still doesn't help, check your work with triangles and geometry...in that way (c) should at least be very straightforward.

4. Oct 28, 2015

### Driverfury

Thanks for your reply. I have not specified it but I want to know the angle between the vectors A and B. For this reason I mentioned the equation: $|A||B|\cos(\theta) =A \cdot B$.

Even if i read your reply I am still not able to evaluate the angle between them (I want a rigorous solution which is valid for every length of S).

And for evaluating I mean that I want to know how many degrees (or radians) is the angle between the two vectors. With no unknown, I want to be able to precisely evalute the angle. In (a) the angle measures 0°, in (b) the angle measures 180°, in (c) it is 90° because 5 = sqrt(a^2 + b^2).

5. Oct 28, 2015

### geoffrey159

Remember that by definition, $s^2 = \vec S.\vec S = (\vec A +\vec B).(\vec A + \vec B)$. You are almost there !

6. Oct 28, 2015

### Staff: Mentor

As you note, parts a) and b) are obvious, and part c) relates to a well-known right triangle.
If the only information you have are the magnitudes of vectors A and B and the magnitude of S, you don't have enough to go on for all possible values of |S|. For example, if |S| = 0 or if |S| > 7, there is no solution.

If S is such that it could be the third side of a triangle, you can use the Law of Cosines to find one of the angles of the triangle formed by A, B, and A + B. With that information, you could find the vector A + B.

7. Oct 28, 2015

### Driverfury

Ok. Thanks for your replies, I solved.