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 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: Define the New Vectors
Let:
\[ \vec{u} = 3(\vec{a} \times \vec{b}), \quad \vec{v} = 4(\vec{b} \times \vec{c}), \quad \vec{w} = 5(\vec{c} \times \vec{a}) \]

### Step 3: Calculate the Volume
Using the volume formula:
\[ V = \frac{1}{6} |\vec{u} \cdot (\vec{v} \times \vec{w})| \]

Substituting the new vectors:
\[ V = \frac{1}{6} |3(\vec{a} \times \vec{b}) \cdot (4(\vec{b} \times \vec{c}) \times 5(\vec{c} \times \vec{a})| \]

### Step 4: Simplify the Cross Product
We can simplify the cross product:
\[ \vec{v} \times \vec{w} = 4(\vec{b} \times \vec{c}) \times 5(\vec{c} \times \vec{a}) = 20((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})) \]

Using the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z})\vec{y} - (\vec{x} \cdot \vec{y})\vec{z} \]
we can rewrite:
\[ (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a} \]

### Step 5: Substitute Back into the Volume Formula
Now substituting back:
\[ V = \frac{1}{6} |3(\vec{a} \times \vec{b}) \cdot (20((\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a}))| \]

### Step 6: Factor Out Constants
This simplifies to:
\[ V = \frac{1}{6} \cdot 60 |(\vec{a} \times \vec{b}) \cdot ((\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a})| \]

### Step 7: Use Given Volume Information
Given that the volume of the tetrahedron formed by \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is \(\frac{3}{4}\) cubic units, we have:
\[ \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| = \frac{3}{4} \]
Thus:
\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = \frac{3}{4} \cdot 6 = \frac{9}{2} \]

### Step 8: Substitute the Volume Expression
Now substituting this back into our volume expression:
\[ V = 10 \cdot \left(\frac{9}{2}\right) = 45 \]

### Final Calculation
Thus the final volume of the tetrahedron determined by the vectors \(3(\vec{a} \times \vec{b})\), \(4(\vec{b} \times \vec{c})\), and \(5(\vec{c} \times \vec{a})\) is:
\[ V = \frac{405}{4} = 101.25 \text{ cubic units} \]