If the angle between the vectors `veca and vecb " is " pi/3` and the area of the triangle with adjacemnt sides parallel to `veca and vecb` is 3 , then a.b is

Options

A. `sqrt(3)`
B. `2sqrt(3)`
C. `4sqrt(3)`
D. `(sqrt(3))/(2)`

Answer

To solve the problem step by step, we will follow the given information and use the relevant formulas.

### Step 1: Understand the given information
We are given:
- The angle between vectors **a** and **b** is \( \theta = \frac{\pi}{3} \).
- The area of the triangle formed by these vectors is 3.

### Step 2: Use the area of a triangle formula
The area \( A \) of a triangle formed by two vectors **a** and **b** can be expressed as:
\[ A = \frac{1}{2} |\mathbf{a} \times \mathbf{b}| \]
Given that the area is 3, we can set up the equation:
\[ 3 = \frac{1}{2} |\mathbf{a} \times \mathbf{b}| \]

### Step 3: Solve for the magnitude of the cross product
Multiplying both sides of the equation by 2 gives:
\[ |\mathbf{a} \times \mathbf{b}| = 6 \]

### Step 4: Relate the cross product to the sine of the angle
The magnitude of the cross product can also be expressed in terms of the magnitudes of the vectors and the sine of the angle between them:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \]
Substituting \( \theta = \frac{\pi}{3} \):
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin\left(\frac{\pi}{3}\right) \]
Since \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), we have:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \cdot \frac{\sqrt{3}}{2} \]

### Step 5: Set the two expressions for the cross product equal
Now we can set the two expressions for the magnitude of the cross product equal to each other:
\[ |\mathbf{a}| |\mathbf{b}| \cdot \frac{\sqrt{3}}{2} = 6 \]

### Step 6: Solve for the product of the magnitudes
Multiplying both sides by \( \frac{2}{\sqrt{3}} \):
\[ |\mathbf{a}| |\mathbf{b}| = 6 \cdot \frac{2}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3} \]

### Step 7: Use the dot product formula
The dot product of two vectors can be expressed as:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]
Substituting \( \theta = \frac{\pi}{3} \) and \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cdot \frac{1}{2} \]

### Step 8: Substitute the value of \( |\mathbf{a}| |\mathbf{b}| \)
Now substituting \( |\mathbf{a}| |\mathbf{b}| = 4\sqrt{3} \):
\[ \mathbf{a} \cdot \mathbf{b} = 4\sqrt{3} \cdot \frac{1}{2} = 2\sqrt{3} \]

### Final Answer
Thus, the value of \( \mathbf{a} \cdot \mathbf{b} \) is \( 2\sqrt{3} \).

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