If `atimes(btimesc)=0`, then

Options

A. `|a|=|b|*|c|=1`
B. `b||c`
C. `a||b`
D. `bdotc`

Answer

To solve the problem, we start with the given equation:

\[ a \times (b \times c) = 0 \]

### Step 1: Use the Vector Triple Product Identity
We can utilize the vector triple product identity, which states that:

\[ a \times (b \times c) = (a \cdot c) b - (a \cdot b) c \]

### Step 2: Set the Expression Equal to Zero
Since we know that \( a \times (b \times c) = 0 \), we can set the expression from the vector triple product identity equal to zero:

\[ (a \cdot c) b - (a \cdot b) c = 0 \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:

\[ (a \cdot c) b = (a \cdot b) c \]

### Step 4: Factor Out Scalars
We can express this as:

\[ (a \cdot c) b - (a \cdot b) c = 0 \]

This implies that the two vectors \( (a \cdot c) b \) and \( (a \cdot b) c \) are equal.

### Step 5: Analyze the Result
This means that the vector \( b \) can be expressed in terms of \( c \) if we divide both sides by \( a \cdot c \) (assuming \( a \cdot c \neq 0 \)):

\[ b = \frac{(a \cdot b)}{(a \cdot c)} c \]

### Step 6: Conclusion
This indicates that \( b \) is a scalar multiple of \( c \), which means that \( b \) is parallel to \( c \).

Thus, the final conclusion is:

\[ \text{b is parallel to c} \]

### Final Answer
The correct option is: **b is parallel to c vector.**

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