Let `veca,vec b,vec c` are three vectors along the adjacent edges ofa tetrahedron, if `|vec a|=|vec b|=|vec c|=2` and `vec a*vec b=vec b*vec c=vec c * vec a=2` then volume of tetrahedron is (A) `1/sqrt2` (B) `2/sqrt3` (C) `sqrt3/2` (D) `2sqrt2/3`

Options

A. `(1)/(sqrt(2))`
B. `(2)/(sqrt(3))`
C. `(sqrt(3))/(2)`
D. `(2sqrt(2))/(3)`

Answer

To find the volume of the tetrahedron formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can use the formula for the volume \(V\) of a tetrahedron given by:

\[ V = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 1: Calculate the magnitudes and dot products

Given:
- \(|\vec{a}| = |\vec{b}| = |\vec{c}| = 2\)
- \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 2\)

We can express the dot products in terms of the magnitudes of the vectors:

\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta_{ab} = 2 \cdot 2 \cdot \cos \theta_{ab} = 4 \cos \theta_{ab} \]

Setting this equal to the given value of 2:

\[ 4 \cos \theta_{ab} = 2 \implies \cos \theta_{ab} = \frac{1}{2} \implies \theta_{ab} = 60^\circ \]

Similarly, we can find that \(\theta_{bc} = 60^\circ\) and \(\theta_{ca} = 60^\circ\).

### Step 2: Calculate the scalar triple product

To find \(|\vec{a} \cdot (\vec{b} \times \vec{c})|\), we can use the formula:

\[ |\vec{b} \times \vec{c}| = |\vec{b}| |\vec{c}| \sin \theta_{bc} \]

Since \(|\vec{b}| = |\vec{c}| = 2\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\):

\[ |\vec{b} \times \vec{c}| = 2 \cdot 2 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \]

Now, we can find \(|\vec{a} \cdot (\vec{b} \times \vec{c})|\):

\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = |\vec{a}| |\vec{b} \times \vec{c}| \cos \theta_{a(b \times c)} \]

Since \(\theta_{a(b \times c)}\) is also \(60^\circ\):

\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = 2 \cdot 2\sqrt{3} \cdot \frac{1}{2} = 2\sqrt{3} \]

### Step 3: Calculate the volume

Now we can substitute this back into the volume formula:

\[ V = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| = \frac{1}{6} (2\sqrt{3}) = \frac{\sqrt{3}}{3} \]

### Step 4: Final answer

The volume of the tetrahedron is:

\[ V = \frac{\sqrt{3}}{3} \]

### Conclusion

The correct answer is:

\[ \text{(C) } \frac{\sqrt{3}}{2} \]