To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables and translating the constraints and objective function into algebraic terms.

Let's define:
- $x_1$ as the number of cups of orange juice used.
- $x_2$ as the number of cups of apple juice used.

The objective is to minimize the amount of Vitamin K in the mix. Given that a cup of orange juice contains 3 units of Vitamin K and a cup of apple juice contains 9 units of Vitamin K, the objective function can be represented as:
\[ \text{Minimize} \quad 3x_1 + 9x_2 \]

The constraints are based on the requirements for Vitamins A, D, and E:
- At least 100 units of Vitamin A: $6x_1 + 10x_2 \geq 100$
- At least 200 units of Vitamin D: $5x_1 + 3x_2 \geq 200$
- At most 150 units of Vitamin E: $12x_1 + 15x_2 \leq 150$

Also, since we cannot have negative cups of juice, we have non-negativity constraints:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'cups of orange juice'), ('x2', 'cups of apple juice')],
    'objective_function': '3*x1 + 9*x2',
    'constraints': ['6*x1 + 10*x2 >= 100', '5*x1 + 3*x2 >= 200', '12*x1 + 15*x2 <= 150', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Vitamin_Mix")

# Add variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="orange_juice")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="apple_juice")

# Set the objective function
m.setObjective(3*x1 + 9*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(6*x1 + 10*x2 >= 100, "Vitamin_A")
m.addConstr(5*x1 + 3*x2 >= 200, "Vitamin_D")
m.addConstr(12*x1 + 15*x2 <= 150, "Vitamin_E")
m.addConstr(x1 >= 0, "Non_Negativity_Orange")
m.addConstr(x2 >= 0, "Non_Negativity_Apple")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Orange Juice: {x1.x}")
    print(f"Apple Juice: {x2.x}")
    print(f"Objective Function Value (Vitamin K): {m.objVal}")
else:
    print("No optimal solution found.")
```