To determine all possible locations of the fourth vertex of the parallelogram, we use the property that opposite sides of a parallelogram are equal and parallel. This implies that the diagonals of the parallelogram bisect each other.

Step 1: Representation of the problem

The given points are:

• $O(0, 0, 0)$
• $P(3, 2, 6)$
• $Q(5, 2, 4)$

Let the fourth vertex be $R(x, y, z)$.

Step 2: Properties of a parallelogram

Using vector addition, if $R$ is the fourth vertex, the two possible parallelogram formations are:

1. $O, P, Q, R$ where $\vec{OR} = \vec{PQ}$.
2. $O, Q, P, R$ where $\vec{OR} = \vec{QP}$.

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Case 1: $\vec{OR} = \vec{PQ}$

The vector $\vec{PQ}$ is: $\vec{PQ} = \vec{Q} - \vec{P} = (5 - 3, 2 - 2, 4 - 6) = (2, 0, -2)$ Thus: $\vec{OR} = (x - 0, y - 0, z - 0) = (2, 0, -2)$ This gives: $R = (2, 0, -2)$

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Case 2: $\vec{OR} = \vec{QP}$

The vector $\vec{QP}$ is: $\vec{QP} = \vec{P} - \vec{Q} = (3 - 5, 2 - 2, 6 - 4) = (-2, 0, 2)$ Thus: $\vec{OR} = (x - 0, y - 0, z - 0) = (-2, 0, 2)$ This gives: $R = (-2, 0, 2)$

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Case 3: Midpoint of diagonals $\vec{OP} + \vec{RQ} = \vec{OQ} + \vec{RP}$

Using the property that the diagonals bisect each other: $\text{Midpoint of } \vec{OP} = \text{Midpoint of } \vec{RQ}$ $\text{Midpoint of } \vec{OQ} = \text{Midpoint of } \vec{RP}$ We confirm that no additional solutions arise.

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Possible locations for the fourth vertex:

From the above, the correct vertices are:

• $(2, 0, -2)$ (Option B)
• $(-2, 0, 2)$ (Option C)

Thus, the correct answers are: B and C.

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Would you like a detailed step-by-step explanation of any part?
Here are some follow-up questions to deepen understanding:

1. How can the diagonals' bisection property help identify parallelograms in other dimensions?
2. Why do parallelograms have exactly two possible solutions for the fourth vertex?
3. How can vector addition and subtraction be used to solve geometric problems in three dimensions?
4. How does this approach generalize to other quadrilaterals like rectangles or rhombuses?
5. What are some real-life applications of vector geometry?

Tip: When solving parallelogram problems, always verify both the sides and diagonals to ensure accuracy.