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

Options

A. `2 : 3`
B. `5 : 7`
C. `15 : 1`
D. `1 : 1`

Answer

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

### Step 1: Define the volumes
The volume \( V_1 \) of the parallelepiped formed by the 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 \( \vec{p}, \vec{q}, \vec{r} \)
We have:
\[ \vec{p} = \vec{a} + \vec{b} - 2\vec{c} \]
\[ \vec{q} = 3\vec{a} - 2\vec{b} + \vec{c} \]
\[ \vec{r} = \vec{a} - 4\vec{b} + 2\vec{c} \]

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

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

Using the distributive property of the cross product:
\[ = 3\vec{a} \times \vec{a} - 12\vec{a} \times \vec{b} + 6\vec{a} \times \vec{c} + 2\vec{b} \times \vec{a} - 8\vec{b} \times \vec{b} + 2\vec{b} \times \vec{c} + \vec{c} \times \vec{a} - 4\vec{c} \times \vec{b} + 2\vec{c} \times \vec{c} \]
Since the cross product of any vector with itself is zero:
\[ = -12\vec{a} \times \vec{b} + 6\vec{a} \times \vec{c} + 2\vec{b} \times \vec{c} + \vec{c} \times \vec{a} - 4\vec{c} \times \vec{b} \]

### Step 5: Simplify \( \vec{q} \times \vec{r} \)
Combining like terms:
\[ = -12\vec{a} \times \vec{b} + 6\vec{a} \times \vec{c} + 2\vec{b} \times \vec{c} - 4\vec{b} \times \vec{c} + \vec{c} \times \vec{a} \]
\[ = -12\vec{a} \times \vec{b} + 6\vec{a} \times \vec{c} - 2\vec{b} \times \vec{c} + \vec{c} \times \vec{a} \]

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

### Step 7: Find the ratio \( V_2 : V_1 \)
After calculating the scalar triple products and simplifying, we find:
\[ V_2 = 15 |\vec{a} \cdot (\vec{b} \times \vec{c})| = 15 V_1 \]
Thus, the ratio is:
\[ \frac{V_2}{V_1} = 15 : 1 \]

### Final Answer
The ratio \( V_2 : V_1 = 15 : 1 \).