The volume of a tetrahedron determined by the vectors `veca, vecb, vecc` is `(3)/(4)` cubic units. The volume (in cubic units) of a tetrahedron determined by the vectors `3(veca xx vecb), 4(vecbxxc) and 5(vecc xx veca)` will be

Answer

To find the volume of the new tetrahedron determined by the vectors \(3(\vec{a} \times \vec{b})\), \(4(\vec{b} \times \vec{c})\), and \(5(\vec{c} \times \vec{a})\), we can follow these steps:

### Step 1: Understand the volume of a tetrahedron
The volume \(V\) of a tetrahedron formed by vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) can be calculated using the formula:
\[ V = \frac{1}{6} |\vec{u} \cdot (\vec{v} \times \vec{w})| \]

### Step 2: Identify the given volume
We know that the volume of the tetrahedron formed by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given as \(\frac{3}{4}\) cubic units. According to the formula:
\[ \frac{3}{4} = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 3: Calculate the scalar triple product
From the equation above, we can solve for the scalar triple product:
\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = \frac{3}{4} \times 6 = \frac{9}{2} \]

### Step 4: Set up the volume for the new tetrahedron
Now, we need to find the volume of the tetrahedron formed by the vectors \(3(\vec{a} \times \vec{b})\), \(4(\vec{b} \times \vec{c})\), and \(5(\vec{c} \times \vec{a})\). Using the volume formula:
\[ V' = \frac{1}{6} |3(\vec{a} \times \vec{b}) \cdot (4(\vec{b} \times \vec{c}) \times 5(\vec{c} \times \vec{a})| \]

### Step 5: Simplify the expression
We can factor out the constants:
\[ V' = \frac{1}{6} \cdot 3 \cdot 4 \cdot 5 \cdot |\vec{a} \times \vec{b} \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}))| \]
Calculating the constants:
\[ 3 \cdot 4 \cdot 5 = 60 \]
Thus,
\[ V' = \frac{60}{6} |\vec{a} \times \vec{b} \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}))| \]
\[ V' = 10 |\vec{a} \times \vec{b} \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}))| \]

### Step 6: Use the vector triple product identity
Using the vector triple product identity:
\[ \vec{x} \cdot (\vec{y} \times \vec{z}) = \vec{y} \cdot (\vec{z} \times \vec{x}) \]
we can express:
\[ (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a})\vec{c} - (\vec{c} \cdot \vec{a})\vec{b} \]
Thus, we can substitute this into our volume expression.

### Step 7: Calculate the final volume
The final volume becomes:
\[ V' = 10 \cdot \frac{9}{2} = 45 \]

### Conclusion
The volume of the new tetrahedron is \(45\) cubic units.

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