Given information:
- A, B, C are unit vectors.
- A•B = A•C = 0.
- The angle between B and C is 30 degrees.

To prove: A = 2(B×C)

Proof:

1. Orthogonality:
Since A•B = 0 and A•C = 0, it implies that vectors A, B, and C are mutually orthogonal (perpendicular) to each other. This can be visualized as vectors B and C lying in a plane perpendicular to vector A.

2. Cross product:
The cross product of vectors B and C, denoted as B×C, gives us a vector that is perpendicular to both B and C. Therefore, B×C is also perpendicular to vector A.

3. Angle between B and C:
The given information states that the angle between vectors B and C is 30 degrees. Since B and C are unit vectors, their magnitude is 1. Therefore, the magnitude of the cross product B×C can be determined using the formula:

|B×C| = |B| |C| sinθ,

where θ is the angle between B and C. Plugging in the values, we have:

|B×C| = (1) (1) sin(30°) = 0.5.

4. Magnitude of A:
The magnitude of vector A is given as 1. Since A, B, and C are mutually orthogonal, the magnitude of the cross product 2(B×C) can be determined as:

|2(B×C)| = 2 |B×C| = 2(0.5) = 1.

5. Direction of A:
The direction of vector A is perpendicular to both B and C. Since B×C is also perpendicular to B and C, the direction of 2(B×C) is the same as that of A.

6. Conclusion:
From the above analysis, it can be concluded that the magnitude and direction of vector A are the same as that of 2(B×C). Therefore, A = 2(B×C) is correct.

Note: It is important to note that the given information regarding the orthogonality of A, B, and C, as well as the angle between B and C, are crucial in proving the equation A = 2(B×C). Without these conditions, it may not hold true.