If `vecp=2veca-3vecb, vecq=veca-2b+vecc, vecr=-3veca+vecb+2vecc, veca,vecb,vecc` being non-zero, non-coplanar vectors, then `-2veca+vecb-vecc` is equal to

Options

A. `vecp-4vecq`
B. `((-7vecq+vecr))/(5)`
C. `2vecp-3vecq+vecr`
D. `4vecp-2vecr`

Answer

To solve the problem, we need to find the expression for \(-2\vec{a} + \vec{b} - \vec{c}\) using the given vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\).

### Given:
1. \(\vec{p} = 2\vec{a} - 3\vec{b}\)
2. \(\vec{q} = \vec{a} - 2\vec{b} + \vec{c}\)
3. \(\vec{r} = -3\vec{a} + \vec{b} + 2\vec{c}\)

### We need to find:
\[ -2\vec{a} + \vec{b} - \vec{c} \]

### Step 1: Check Option 1: \(\vec{p} - 4\vec{q}\)
\[ \vec{p} - 4\vec{q} = (2\vec{a} - 3\vec{b}) - 4(\vec{a} - 2\vec{b} + \vec{c}) \]
Expanding this:
\[ = 2\vec{a} - 3\vec{b} - 4\vec{a} + 8\vec{b} - 4\vec{c} \]
Combining like terms:
\[ = (2\vec{a} - 4\vec{a}) + (-3\vec{b} + 8\vec{b}) - 4\vec{c} = -2\vec{a} + 5\vec{b} - 4\vec{c} \]
This is not equal to \(-2\vec{a} + \vec{b} - \vec{c}\).

### Step 2: Check Option 2: \(-7\vec{q} + \vec{r}\)
\[ -7\vec{q} + \vec{r} = -7(\vec{a} - 2\vec{b} + \vec{c}) + (-3\vec{a} + \vec{b} + 2\vec{c}) \]
Expanding:
\[ = -7\vec{a} + 14\vec{b} - 7\vec{c} - 3\vec{a} + \vec{b} + 2\vec{c} \]
Combining like terms:
\[ = (-7\vec{a} - 3\vec{a}) + (14\vec{b} + \vec{b}) + (-7\vec{c} + 2\vec{c}) = -10\vec{a} + 15\vec{b} - 5\vec{c} \]
This is not equal to \(-2\vec{a} + \vec{b} - \vec{c}\).

### Step 3: Check Option 3: \(2\vec{p} - 3\vec{q} + \vec{r}\)
\[ 2\vec{p} - 3\vec{q} + \vec{r} = 2(2\vec{a} - 3\vec{b}) - 3(\vec{a} - 2\vec{b} + \vec{c}) + (-3\vec{a} + \vec{b} + 2\vec{c}) \]
Expanding:
\[ = 4\vec{a} - 6\vec{b} - 3\vec{a} + 6\vec{b} - 3\vec{c} - 3\vec{a} + \vec{b} + 2\vec{c} \]
Combining like terms:
\[ = (4\vec{a} - 3\vec{a} - 3\vec{a}) + (-6\vec{b} + 6\vec{b} + \vec{b}) + (-3\vec{c} + 2\vec{c}) = -2\vec{a} + \vec{b} - \vec{c} \]
This is equal to \(-2\vec{a} + \vec{b} - \vec{c}\).

### Step 4: Check Option 4: \(4\vec{p} - 2\vec{r}\)
\[ 4\vec{p} - 2\vec{r} = 4(2\vec{a} - 3\vec{b}) - 2(-3\vec{a} + \vec{b} + 2\vec{c}) \]
Expanding:
\[ = 8\vec{a} - 12\vec{b} + 6\vec{a} - 2\vec{b} - 4\vec{c} \]
Combining like terms:
\[ = (8\vec{a} + 6\vec{a}) + (-12\vec{b} - 2\vec{b}) - 4\vec{c} = 14\vec{a} - 14\vec{b} - 4\vec{c} \]
This is not equal to \(-2\vec{a} + \vec{b} - \vec{c}\).

### Conclusion:
The only option that satisfies the equation \(-2\vec{a} + \vec{b} - \vec{c}\) is:
\[ \boxed{2\vec{p} - 3\vec{q} + \vec{r}} \]