If `veca=2hati-3hatj+5hatk , vecb=3hati-4hatj+5hatk` and `vecc=5hati-3hatj-2hatk`, then the volume of the parallelopiped with coterminous edges `veca+vecb,vecb+vecc,vecc+veca` is

Options

A. 2
B. 1
C. 16
D. 0

Explanation

To find the volume of the parallelepiped formed by the vectors \(\vec{a} + \vec{b}\), \(\vec{b} + \vec{c}\), and \(\vec{c} + \vec{a}\), we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \vec{a} = 2\hat{i} - 3\hat{j} + 5\hat{k} \]
\[ \vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k} \]
\[ \vec{c} = 5\hat{i} - 3\hat{j} - 2\hat{k} \]

### Step 2: Calculate \(\vec{p} = \vec{a} + \vec{b}\)
\[ \vec{p} = \vec{a} + \vec{b} = (2 + 3)\hat{i} + (-3 - 4)\hat{j} + (5 + 5)\hat{k} \]
\[ \vec{p} = 5\hat{i} - 7\hat{j} + 10\hat{k} \]

### Step 3: Calculate \(\vec{q} = \vec{b} + \vec{c}\)
\[ \vec{q} = \vec{b} + \vec{c} = (3 + 5)\hat{i} + (-4 - 3)\hat{j} + (5 - 2)\hat{k} \]
\[ \vec{q} = 8\hat{i} - 7\hat{j} + 3\hat{k} \]

### Step 4: Calculate \(\vec{r} = \vec{c} + \vec{a}\)
\[ \vec{r} = \vec{c} + \vec{a} = (5 + 2)\hat{i} + (-3 - 3)\hat{j} + (-2 + 5)\hat{k} \]
\[ \vec{r} = 7\hat{i} - 6\hat{j} + 3\hat{k} \]

### Step 5: Set up the determinant for volume
The volume \(V\) of the parallelepiped formed by vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) is given by the absolute value of the determinant of the matrix formed by these vectors:
\[ V = \left| \begin{vmatrix} 5 & -7 & 10 \\ 8 & -7 & 3 \\ 7 & -6 & 3 \end{vmatrix} \right| \]

### Step 6: Calculate the determinant
Calculating the determinant:
\[ \begin{vmatrix} 5 & -7 & 10 \\ 8 & -7 & 3 \\ 7 & -6 & 3 \end{vmatrix} = 5 \begin{vmatrix} -7 & 3 \\ -6 & 3 \end{vmatrix} - (-7) \begin{vmatrix} 8 & 3 \\ 7 & 3 \end{vmatrix} + 10 \begin{vmatrix} 8 & -7 \\ 7 & -6 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \(\begin{vmatrix} -7 & 3 \\ -6 & 3 \end{vmatrix} = (-7)(3) - (3)(-6) = -21 + 18 = -3\)

2. \(\begin{vmatrix} 8 & 3 \\ 7 & 3 \end{vmatrix} = (8)(3) - (3)(7) = 24 - 21 = 3\)

3. \(\begin{vmatrix} 8 & -7 \\ 7 & -6 \end{vmatrix} = (8)(-6) - (-7)(7) = -48 + 49 = 1\)

Substituting back into the determinant:
\[ = 5(-3) + 7(3) + 10(1) = -15 + 21 + 10 = 16 \]

### Step 7: Final volume
Thus, the volume of the parallelepiped is:
\[ V = |16| = 16 \]

### Conclusion
The volume of the parallelepiped is \(16\).

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