If `|veca|=2 and |vecb|=3 and veca.vecb=0, " then " (vecaxx(vecaxx(vecaxx(vecaxxvecb))))` is equal to the given diagonal is `vecc = 4 hatk = 8hatk` then , the volume of a parallelpiped is

Options

A. `48 hatb`
B. `-48 hatb`
C. `48 hata`
D. `-48 hata`

Explanation

To solve the problem, we need to find the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). The volume \(V\) of a parallelepiped defined by three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) can be calculated using the scalar triple product:

\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

Given:
- \(|\vec{a}| = 2\)
- \(|\vec{b}| = 3\)
- \(\vec{a} \cdot \vec{b} = 0\) (which means \(\vec{a}\) and \(\vec{b}\) are perpendicular)

We also have the expression for \(\vec{a} \times \vec{a} \times \vec{a} \times \vec{a} \times \vec{b}\). Since the cross product of any vector with itself is zero, we can simplify the expression.

### Step 1: Simplify the expression
\[ \vec{a} \times \vec{a} = \vec{0} \]
Thus,
\[ \vec{a} \times \vec{a} \times \vec{a} \times \vec{a} \times \vec{b} = \vec{0} \times \vec{b} = \vec{0} \]

### Step 2: Find the vector \(\vec{c}\)
We are given that the diagonal vector \(\vec{c} = 4\hat{k} + 8\hat{k} = 12\hat{k}\).

### Step 3: Calculate the volume
The volume of the parallelepiped can be calculated using the formula:
\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 4: Find \(\vec{b} \times \vec{c}\)
Since \(\vec{b}\) is perpendicular to \(\vec{a}\), we can assume:
\[ \vec{b} = 3\hat{j} \]
And \(\vec{c} = 12\hat{k}\).

Now, we calculate:
\[ \vec{b} \times \vec{c} = (3\hat{j}) \times (12\hat{k}) = 36\hat{i} \]

### Step 5: Calculate \(\vec{a} \cdot (\vec{b} \times \vec{c})\)
Assuming \(\vec{a} = 2\hat{i}\):
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = (2\hat{i}) \cdot (36\hat{i}) = 72 \]

### Step 6: Find the volume
Thus, the volume \(V\) is:
\[ V = |72| = 72 \]

### Final Answer
The volume of the parallelepiped is \(72\) cubic units.

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