Explanation:
Given that A vector dot B vector =0 and A vector dot C vector =0.

We need to determine the possible directions of A vector.

Option 1: A vector can be parallel to C vector
If A vector is parallel to C vector, then A vector dot C vector will not be equal to zero. Hence, this option is not correct.

Option 2: A vector can be parallel to B vector
If A vector is parallel to B vector, then A vector dot B vector will not be equal to zero. Hence, this option is not correct.

Option 3: A vector can be parallel to B vector cross C vector
Let's assume that A vector is parallel to B vector cross C vector. Then, we have:
A vector = k(B vector cross C vector), where k is a constant.

Now, A vector dot B vector = 0 implies that:
k(B vector cross C vector) dot B vector = 0
k(B vector dot B vector) cross (C vector dot B vector) = 0
k(B vector dot B vector)(C vector dot B vector)sin(theta) = 0, where theta is the angle between B vector and C vector.

Since B vector dot B vector is not equal to zero, we have:
(C vector dot B vector)sin(theta) = 0
This implies that either C vector is parallel to B vector or they are perpendicular to each other.

Similarly, A vector dot C vector = 0 implies that either C vector is parallel to B vector or they are perpendicular to each other.

Hence, if A vector is parallel to B vector cross C vector, then C vector must be parallel to B vector or they must be perpendicular to each other. But this is not always true. Hence, this option is also not correct.

Option 4: A vector can be perpendicular to both B vector and C vector
Let's assume that A vector is perpendicular to both B vector and C vector. Then, we have:
A vector dot B vector = 0 and A vector dot C vector = 0.

Hence, this option is correct.

Conclusion:
Therefore, we can conclude that A vector can be perpendicular to both B vector and C vector.