How to find one of the vectors when we know the other, the angle between the vectors and the result? [closed]

Question

We're given that the result of the two vectors $a$ and $5$ is $7$ and the angle between the two vectors $5$ and $a$ is $60$ $degrees$. How do we calculate the of the vector $a$ ?

I used $R^2 = P^2 + Q^2 + 2PQ\cos \theta$ and go two answers for a $a$. How do I pick the correct one? What is the logic used to pick the correct one? $3$ or $-8$

$R^2 = P^2 + Q^2 + 2PQ\cos \frac \pi3 $

$49 = a^2 + 25 + 5a$

$a^2 + 5a - 24 = 0$

$(a-3)(a+8) = 0$

$ a= 3$ or $a = -8$

Answer 1

both are correct since both of them satisfy the equation