Three vectors `a(|a|ne0), b` and c are such that `atimesb=3atimesc`, also `|a|=|b|=1 and |c|=(1)/(3)`. If the angle between b and c is `60^(@)` and `|b-3x|=lambda|a|`, then the value of `lambda` is

Answer

To solve the problem, we need to analyze the given vectors and their properties step by step.

### Given:
1. \( \mathbf{a} \times \mathbf{b} = 3(\mathbf{a} \times \mathbf{c}) \)
2. \( |\mathbf{a}| = |\mathbf{b}| = 1 \)
3. \( |\mathbf{c}| = \frac{1}{3} \)
4. The angle between \( \mathbf{b} \) and \( \mathbf{c} \) is \( 60^\circ \).
5. \( |\mathbf{b} - 3\mathbf{c}| = \lambda |\mathbf{a}| \)

### Step 1: Square both sides of the equation \( |\mathbf{b} - 3\mathbf{c}| = \lambda |\mathbf{a}| \)

\[ |\mathbf{b} - 3\mathbf{c}|^2 = \lambda^2 |\mathbf{a}|^2 \]

Since \( |\mathbf{a}| = 1 \):

\[ |\mathbf{b} - 3\mathbf{c}|^2 = \lambda^2 \]

### Step 2: Expand the left-hand side

Using the formula \( | \mathbf{u} - \mathbf{v} |^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2 \mathbf{u} \cdot \mathbf{v} \):

\[ |\mathbf{b}|^2 + |3\mathbf{c}|^2 - 2 \mathbf{b} \cdot (3\mathbf{c}) = \lambda^2 \]

### Step 3: Substitute the magnitudes

We know \( |\mathbf{b}| = 1 \) and \( |3\mathbf{c}| = 3|\mathbf{c}| = 3 \cdot \frac{1}{3} = 1 \):

\[ 1^2 + 1^2 - 2 \cdot 3 \mathbf{b} \cdot \mathbf{c} = \lambda^2 \]

This simplifies to:

\[ 1 + 1 - 6 \mathbf{b} \cdot \mathbf{c} = \lambda^2 \]

### Step 4: Calculate \( \mathbf{b} \cdot \mathbf{c} \)

Using the dot product formula:

\[ \mathbf{b} \cdot \mathbf{c} = |\mathbf{b}| |\mathbf{c}| \cos(60^\circ) \]

Substituting the values:

\[ \mathbf{b} \cdot \mathbf{c} = 1 \cdot \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6} \]

### Step 5: Substitute \( \mathbf{b} \cdot \mathbf{c} \) back into the equation

Now substituting back into the equation:

\[ 2 - 6 \cdot \frac{1}{6} = \lambda^2 \]

This simplifies to:

\[ 2 - 1 = \lambda^2 \]

Thus:

\[ \lambda^2 = 1 \]

### Step 6: Solve for \( \lambda \)

Taking the square root of both sides gives:

\[ \lambda = 1 \quad \text{(since } \lambda \text{ must be positive)} \]

### Final Answer:
\[ \lambda = 1 \]