If I have $3$ vectors, $u,v,w$, and $|u|=6,|v|=9,|w|=12$, how do I write w as a linear combination of u and v? (Just in case the diagram isn't clear, the angle between vector u and v is 30 degrees and the angle between vector w and vector v is 60 degrees)
Diagram depicting situation is below.
Here's a hint.
Think of $\vec{u} = 6 \hat{x}$ and $\vec{v} = -9\hat{y}$, where $\hat{x}, \hat{y}$ are the unit vectors in the $x$ and $y$ directions. (This you can do because $\vec{u} \perp \vec{v}$.)
Then, $\vec{w} = 12 \cos (30^{\circ})\hat{x} - 12 \sin (30^{\circ})\hat{y} = A(6\hat{x}) - B(9\hat{y}) = A\vec{u} + B\vec{v}.$
Then,
$$6A = 12 \cos (30^{\circ}) \\ 9B = 12 \sin (30^{\circ})$$
gives you $A,B$ and from there $\vec{w} = A\vec{u} + B\vec{v}$, which is what you need.