If `veca and vec b` are non-zero, non-collinear vectors such that `|veca|=2, veca*vecb=1` and angle between `veca and vec b` is `(pi)/(3)`. If `vecr` is any vector such that `vecr*veca=2, vecr*vecb=8, (vec r+2veca-10vecb)*(vecaxxvecb)=4sqrt(3)` and satisfy to `vecr+2veca-10vecb=lambda(veca xx vec b)`, then `lambda` is equal to : (a) `1/2` (b) 2 (c) `1/4` (d) none of these

Options

A. `(1)/(2)`
B. 2
C. `(1)/(4)`
D. None of these

Answer

To solve the problem step by step, we will follow the information given in the question and apply the relevant mathematical concepts.

### Step 1: Determine the Magnitude of Vector B
We know that:
- \( |\vec{a}| = 2 \)
- \( \vec{a} \cdot \vec{b} = 1 \)
- The angle \( \theta \) between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \).

Using the dot product formula:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
Substituting the known values:
\[ 1 = 2 |\vec{b}| \cos\left(\frac{\pi}{3}\right) \]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ 1 = 2 |\vec{b}| \cdot \frac{1}{2} \]
This simplifies to:
\[ 1 = |\vec{b}| \]
Thus, the magnitude of vector \( \vec{b} \) is \( 1 \).

### Step 2: Analyze the Vector \( \vec{r} \)
We are given:
- \( \vec{r} \cdot \vec{a} = 2 \)
- \( \vec{r} \cdot \vec{b} = 8 \)

### Step 3: Set Up the Equation for \( \vec{r} \)
We also have:
\[ \vec{r} + 2\vec{a} - 10\vec{b} = \lambda (\vec{a} \times \vec{b}) \]
Taking the dot product of both sides with \( \vec{a} \times \vec{b} \):
\[ (\vec{r} + 2\vec{a} - 10\vec{b}) \cdot (\vec{a} \times \vec{b}) = \lambda (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b}) \]

### Step 4: Calculate \( (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b}) \)
The magnitude of \( \vec{a} \times \vec{b} \) is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta) \]
Substituting the known values:
\[ |\vec{a} \times \vec{b}| = 2 \cdot 1 \cdot \sin\left(\frac{\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]
Thus:
\[ |\vec{a} \times \vec{b}|^2 = 3 \]

### Step 5: Substitute and Solve for \( \lambda \)
Now substituting back into our equation:
\[ (\vec{r} + 2\vec{a} - 10\vec{b}) \cdot (\vec{a} \times \vec{b}) = \lambda \cdot 3 \]
We also know:
\[ \vec{r} + 2\vec{a} - 10\vec{b} \cdot (\vec{a} \times \vec{b}) = 4\sqrt{3} \]
So we have:
\[ 4\sqrt{3} = \lambda \cdot 3 \]
Thus:
\[ \lambda = \frac{4\sqrt{3}}{3} \]

### Step 6: Check the Options
The calculated value of \( \lambda \) is \( \frac{4\sqrt{3}}{3} \). Since this does not match any of the provided options, we conclude that the answer is:
**(d) none of these.**