If a, b and c are any three non-zero vectors, then the component of `atimes(btimesc)` perpendicular to b is

Options

A. `atimes(btimesc)+((atimesb)*(ctimesa))/(|b|^(2))b`
B. `atimes(btimesc)+((atimesc)*(atimesb))/(|b|^(2))b`
C. `atimes(btimesc)+((atimesb)*(btimesa))/(|b|^(2))b`
D. `atimes(btimesc)+((atimesb)*(btimesc))/(|b|^(2))b`

Answer

To solve the problem of finding the component of \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) that is perpendicular to \( \mathbf{b} \), we can follow these steps:

### Step 1: Identify the expression
We start with the expression \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \). We need to find the component of this vector that is perpendicular to \( \mathbf{b} \).

### Step 2: Use the vector triple product identity
We can use the vector triple product identity, which states:
\[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \]
Applying this identity, we can rewrite \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) as:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

### Step 3: Find the component perpendicular to \( \mathbf{b} \)
To find the component of \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) that is perpendicular to \( \mathbf{b} \), we can denote this component as \( \mathbf{x} \). The component of any vector \( \mathbf{v} \) along \( \mathbf{b} \) can be found using the formula:
\[ \text{component of } \mathbf{v} \text{ along } \mathbf{b} = \frac{\mathbf{v} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]
Thus, the component of \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) along \( \mathbf{b} \) is:
\[ \mathbf{y} = \frac{(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})) \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]

### Step 4: Calculate the perpendicular component
The component \( \mathbf{x} \) that we are looking for is given by:
\[ \mathbf{x} = \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) - \mathbf{y} \]
Substituting for \( \mathbf{y} \):
\[ \mathbf{x} = \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) - \frac{(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{b})}{|\mathbf{b}|^2} \mathbf{b} \]

### Step 5: Final expression
Thus, the component of \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) that is perpendicular to \( \mathbf{b} \) is:
\[ \mathbf{x} = \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) - \frac{(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{b})}{|\mathbf{b}|^2} \mathbf{b} \]