(3)

To solve this problem, we will use the concepts of vector addition and magnitudes involving the dot product. Given the angle between and is , we can use the properties of dot products and magnitudes to find the required integer value of .
First, let's use the condition . Square both sides to remove the square roots:

Now we will expand both sides using the formula for the magnitude of the sum and difference of vectors:

And

Substitute these back into the original equation:

Expand the right-hand side:

Rearrange all terms to one side to combine like terms:

0

Combine like terms:

We know that:

Since the angle between and is , we have:

Substitute these back into the equation:

Simplify it further:

We know that . Substitute this into the equation:

Simplify it:

Factor out :

Since , we can solve:

Multiply through by 3 to clear the fraction:

Rearrange it to match standard quadratic form:

Solve this quadratic equation using the quadratic formula:

Here, , and :

This results in two possible solutions for :

However, since is given to be an integer, we take: