Let `veca, vecb, vecc` be three non-zero non coplanar vectors and `vecp, vecq` and `vecr` be three vectors given by `vecp=veca+vecb-2vecc, vecq=3veca-2vecb+vecc` and `vecr=veca-4vcb+2vecc`
If the volume of the parallelopiped determined by `veca, vecb` and `vecc` is `V_(1)` and that of the parallelopiped determined by `veca, vecq` and `vecr` is `V_(2)`, then `V_(2):V_(1)=`

Options

A. 10
B. 15
C. 20
D. None of these

Answer

To solve the problem, we need to find the ratio \( V_2 : V_1 \) where \( V_1 \) is the volume of the parallelepiped formed by vectors \( \vec{a}, \vec{b}, \vec{c} \) and \( V_2 \) is the volume formed by vectors \( \vec{a}, \vec{q}, \vec{r} \).

### Step 1: Define the volumes
The volume \( V_1 \) of the parallelepiped formed by vectors \( \vec{a}, \vec{b}, \vec{c} \) is given by the scalar triple product:
\[ V_1 = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 2: Express vectors \( \vec{p}, \vec{q}, \vec{r} \)
Given:
\[ \vec{p} = \vec{a} + \vec{b} - 2\vec{c} \]
\[ \vec{q} = 3\vec{a} - 2\vec{b} + \vec{c} \]
\[ \vec{r} = \vec{a} - 4\vec{b} + 2\vec{c} \]

### Step 3: Calculate \( V_2 \)
The volume \( V_2 \) is given by the scalar triple product:
\[ V_2 = |\vec{p} \cdot (\vec{q} \times \vec{r})| \]

### Step 4: Calculate \( \vec{q} \times \vec{r} \)
First, we need to compute \( \vec{q} \times \vec{r} \):
\[ \vec{q} \times \vec{r} = (3\vec{a} - 2\vec{b} + \vec{c}) \times (\vec{a} - 4\vec{b} + 2\vec{c}) \]

Using the distributive property of the cross product:
\[ \vec{q} \times \vec{r} = 3\vec{a} \times \vec{a} + 12\vec{a} \times \vec{b} - 6\vec{a} \times \vec{c} - 2\vec{b} \times \vec{a} + 8\vec{b} \times \vec{b} - 2\vec{b} \times \vec{c} + \vec{c} \times \vec{a} - 4\vec{c} \times \vec{b} + 2\vec{c} \times \vec{c} \]

Since the cross product of any vector with itself is zero:
\[ \vec{q} \times \vec{r} = 12\vec{a} \times \vec{b} - 6\vec{a} \times \vec{c} + 2\vec{b} \times \vec{c} + \vec{c} \times \vec{a} - 4\vec{c} \times \vec{b} \]

### Step 5: Substitute back into \( V_2 \)
Now substituting into \( V_2 \):
\[ V_2 = |\vec{p} \cdot (\vec{q} \times \vec{r})| = |(\vec{a} + \vec{b} - 2\vec{c}) \cdot (12\vec{a} \times \vec{b} - 6\vec{a} \times \vec{c} + 2\vec{b} \times \vec{c})| \]

### Step 6: Expand the dot product
Expanding this dot product:
\[ V_2 = |(\vec{a} \cdot (12\vec{a} \times \vec{b})) + (\vec{b} \cdot (12\vec{a} \times \vec{b})) - 2(\vec{c} \cdot (12\vec{a} \times \vec{b})) - (\vec{a} \cdot (6\vec{a} \times \vec{c})) - (\vec{b} \cdot (6\vec{a} \times \vec{c})) + 2(\vec{c} \cdot (6\vec{a} \times \vec{c})) + (\vec{a} \cdot (2\vec{b} \times \vec{c})) + (\vec{b} \cdot (2\vec{b} \times \vec{c})) - 2(\vec{c} \cdot (2\vec{b} \times \vec{c}))| \]

Using properties of the scalar triple product, we simplify:
\[ V_2 = |12(\vec{a} \cdot (\vec{b} \times \vec{c})) + 6(\vec{a} \cdot (\vec{b} \times \vec{c})) + 2(\vec{a} \cdot (\vec{b} \times \vec{c}))| \]
\[ = |20(\vec{a} \cdot (\vec{b} \times \vec{c}))| = 20V_1 \]

### Step 7: Find the ratio \( V_2 : V_1 \)
Thus, we find:
\[ \frac{V_2}{V_1} = 20 \]

### Final Answer
\[ V_2 : V_1 = 20 : 1 \]