We are given three unit vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$, along with information about the angles between them, and that these vectors are linearly dependent. Our goal is to find the values of $\frac{a}{c}$ and $\frac{b}{c}$ such that $a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}$, meaning the vectors form a linear combination that results in the zero vector.

Step 1: Parametrize the unit vectors

Since all the vectors are unit vectors, we can express them in terms of their components using the angles provided:

• $\mathbf{u} = (1, 0)$ since it points in the direction of the positive $x$-axis.
• The angle between $\mathbf{u}$ and $\mathbf{v}$ is $150^\circ$, so we can write $\mathbf{v}$ as: $\mathbf{v} = (\cos 150^\circ, \sin 150^\circ) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right).$
• The angle between $\mathbf{v}$ and $\mathbf{w}$ is $120^\circ$, and since $\mathbf{w}$ is a unit vector, we can write it as: $\mathbf{w} = (\cos 110^\circ, \sin 110^\circ).$ However, we don't need to calculate this exactly as we will use linear dependence relations.

Step 2: Use linear dependence

Since $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are linearly dependent, there exist scalars $a$, $b$, and $c$ such that: $a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}.$ This gives us the system of two equations (one for each component):

1. $a + b \cos 150^\circ + c \cos 110^\circ = 0$
2. $b \sin 150^\circ + c \sin 110^\circ = 0$

Solve for ans