If `veca, vecb and vecc` are three vectors such that `3veca+4vecb+6vecc=vec0, |veca|=3, |vecb|=3 and |vecc|=4`, then the value of `-864((veca.vecb+vecb.vecc+vecc.veca)/(6))` is equal to

Answer

To solve the problem, we need to find the value of \(-864 \left( \frac{\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}}{6} \right)\) given the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) satisfy the equation \(3\vec{a} + 4\vec{b} + 6\vec{c} = \vec{0}\) and their magnitudes \(|\vec{a}| = 3\), \(|\vec{b}| = 3\), and \(|\vec{c}| = 4\).

### Step 1: Rearranging the Vector Equation
From the equation \(3\vec{a} + 4\vec{b} + 6\vec{c} = \vec{0}\), we can express one vector in terms of the others:
\[ 3\vec{a} + 4\vec{b} = -6\vec{c} \]

### Step 2: Squaring Both Sides
Now, we square both sides:
\[ (3\vec{a} + 4\vec{b}) \cdot (3\vec{a} + 4\vec{b}) = (-6\vec{c}) \cdot (-6\vec{c}) \]
This gives us:
\[ 9\vec{a} \cdot \vec{a} + 24\vec{a} \cdot \vec{b} + 16\vec{b} \cdot \vec{b} = 36\vec{c} \cdot \vec{c} \]

### Step 3: Substituting Magnitudes
We substitute the magnitudes:
\[ 9|\vec{a}|^2 + 24\vec{a} \cdot \vec{b} + 16|\vec{b}|^2 = 36|\vec{c}|^2 \]
Substituting \(|\vec{a}| = 3\), \(|\vec{b}| = 3\), and \(|\vec{c}| = 4\):
\[ 9(3^2) + 24\vec{a} \cdot \vec{b} + 16(3^2) = 36(4^2) \]
This simplifies to:
\[ 81 + 24\vec{a} \cdot \vec{b} + 144 = 576 \]
Combining terms:
\[ 225 + 24\vec{a} \cdot \vec{b} = 576 \]
Thus:
\[ 24\vec{a} \cdot \vec{b} = 576 - 225 = 351 \]
So:
\[ \vec{a} \cdot \vec{b} = \frac{351}{24} = \frac{117}{8} \]

### Step 4: Finding \(\vec{b} \cdot \vec{c}\)
Using a similar approach, we can express \(4\vec{b} + 6\vec{c} = -3\vec{a}\) and square it:
\[ (4\vec{b} + 6\vec{c}) \cdot (4\vec{b} + 6\vec{c}) = (-3\vec{a}) \cdot (-3\vec{a}) \]
This gives:
\[ 16\vec{b} \cdot \vec{b} + 48\vec{b} \cdot \vec{c} + 36\vec{c} \cdot \vec{c} = 9\vec{a} \cdot \vec{a} \]
Substituting magnitudes:
\[ 16(3^2) + 48\vec{b} \cdot \vec{c} + 36(4^2) = 9(3^2) \]
This simplifies to:
\[ 144 + 48\vec{b} \cdot \vec{c} + 576 = 81 \]
Combining terms:
\[ 720 + 48\vec{b} \cdot \vec{c} = 81 \]
Thus:
\[ 48\vec{b} \cdot \vec{c} = 81 - 720 = -639 \]
So:
\[ \vec{b} \cdot \vec{c} = \frac{-639}{48} \]

### Step 5: Finding \(\vec{c} \cdot \vec{a}\)
Using \(3\vec{a} + 6\vec{c} = -4\vec{b}\) and squaring:
\[ (3\vec{a} + 6\vec{c}) \cdot (3\vec{a} + 6\vec{c}) = (-4\vec{b}) \cdot (-4\vec{b}) \]
This gives:
\[ 9\vec{a} \cdot \vec{a} + 36\vec{a} \cdot \vec{c} + 36\vec{c} \cdot \vec{c} = 16\vec{b} \cdot \vec{b} \]
Substituting magnitudes:
\[ 9(3^2) + 36\vec{a} \cdot \vec{c} + 36(4^2) = 16(3^2) \]
This simplifies to:
\[ 81 + 36\vec{a} \cdot \vec{c} + 576 = 144 \]
Combining terms:
\[ 657 + 36\vec{a} \cdot \vec{c} = 144 \]
Thus:
\[ 36\vec{a} \cdot \vec{c} = 144 - 657 = -513 \]
So:
\[ \vec{a} \cdot \vec{c} = \frac{-513}{36} \]

### Step 6: Summing the Dot Products
Now we sum:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = \frac{117}{8} + \frac{-639}{48} + \frac{-513}{36} \]
Finding a common denominator (which is 144):
\[ \frac{117 \times 18}{144} + \frac{-639 \times 3}{144} + \frac{-513 \times 4}{144} \]
Calculating:
\[ \frac{2106 - 1917 - 2052}{144} = \frac{-1863}{144} \]

### Step 7: Final Calculation
Now we substitute back into the original expression:
\[ -864 \left( \frac{-1863/144}{6} \right) = -864 \left( \frac{-1863}{864} \right) = 1863 \]

### Final Answer
Thus, the value is:
\[ \boxed{1863} \]