1. Interpret the geometry carefully

Let

be unit vectors along the directions of sides

respectively.

Since these are directed sides of triangle $PQR$ taken cyclically, we have

Therefore, for the corresponding unit vectors,

for some positive scalars $a,b,c$ (the side lengths). Hence the three directions are exactly the directions of the three sides of a triangle taken in order.

Let the angles between these unit vectors be related to the triangle angles $P,Q,R$.

2. Find the pairwise angles between the directions

• $\vec{OY}$ is along $RP$, and $\vec{OZ}$ is along $PQ$. The angle between $RP$ and $PQ$ is $$\pi - P$$ because the interior angle at $P$ is between $PR$ and $PQ$, while $RP$ is opposite to $PR$.

So,

Similarly,

and

3. Rewrite the required expression

Since in a triangle,

we get

Therefore,

Using $\cos(\pi-\theta)=-\cos\theta$,

So minimizing $S$ is equivalent to maximizing

4. Use the standard identity for a triangle

For angles of a triangle,

because

and equality holds for an equilateral triangle.

Hence,

So,

Thus the minimum value of $S$ is

5. Check equality case

For an equilateral triangle,

Then

So the bound is attained.

6. Evaluate options

• A: $-\dfrac{3}{2}$ ✅
• B: $\dfrac{3}{2}$ ❌
• C: $\dfrac{5}{3}$ ❌
• D: $-\dfrac{5}{3}$ ❌

Therefore, the correct option is A.