Three vectors satisfy the relation A.B =0 and A.C=0 then A is parallel to

Options

A. B
B. C
C. B.C
D. `BxxC`

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the conclusion logically.

### Step 1: Understand the Given Conditions
We are given two conditions involving three vectors A, B, and C:
1. \( A \cdot B = 0 \)
2. \( A \cdot C = 0 \)

### Step 2: Analyze the First Condition
The first condition \( A \cdot B = 0 \) indicates that the dot product of vectors A and B is zero. The dot product can be expressed as:
\[ A \cdot B = |A| |B| \cos \theta \]
where \( \theta \) is the angle between vectors A and B. Since the dot product is zero and both A and B are non-zero vectors, we conclude that:
\[ \cos \theta = 0 \]
This implies:
\[ \theta = 90^\circ \]
Thus, vector A is perpendicular to vector B.

### Step 3: Analyze the Second Condition
Similarly, for the second condition \( A \cdot C = 0 \), we apply the same reasoning:
\[ A \cdot C = |A| |C| \cos \phi \]
where \( \phi \) is the angle between vectors A and C. Since this dot product is also zero, we have:
\[ \cos \phi = 0 \]
This implies:
\[ \phi = 90^\circ \]
Thus, vector A is also perpendicular to vector C.

### Step 4: Conclusion About the Relationship of A with B and C
Now we have established that:
- A is perpendicular to B
- A is perpendicular to C

### Step 5: Determine the Direction of A
Since A is perpendicular to both B and C, we can find a vector that is perpendicular to both B and C. This is given by the cross product:
\[ B \times C \]
The vector \( B \times C \) is perpendicular to the plane formed by vectors B and C.

### Step 6: Establish the Parallel Relationship
Since vector A is perpendicular to both B and C, it must be parallel to the vector \( B \times C \). Therefore, we conclude that:
\[ A \text{ is parallel to } (B \times C) \]

### Final Answer
Thus, the answer to the question is that vector A is parallel to the vector \( B \times C \).

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