If two diagonals of one of its faces are ` 6hati + 6 hatk and 4 hatj + 2hatk` and of the edges not containing the given diagonals is `vecc= 4 hatj - 8 hatk` , then the volume of a parallelpiped is

Options

A. 60
B. 80
C. 100
D. 120

Explanation

To find the volume of the parallelepiped given the diagonals of one of its faces and an edge not containing those diagonals, we can follow these steps:

### Step 1: Define the vectors
Let:
- \( \vec{a} = 6\hat{i} + 0\hat{j} + 6\hat{k} \)
- \( \vec{b} = 0\hat{i} + 4\hat{j} + 2\hat{k} \)
- \( \vec{c} = 0\hat{i} + 4\hat{j} - 8\hat{k} \)

### Step 2: Calculate the cross product \( \vec{a} \times \vec{b} \)
Using the determinant method:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 0 & 6 \\ 0 & 4 & 2 \end{vmatrix} \]

Calculating the determinant:
\[ \vec{a} \times \vec{b} = \hat{i}(0 \cdot 2 - 6 \cdot 4) - \hat{j}(6 \cdot 2 - 6 \cdot 0) + \hat{k}(6 \cdot 4 - 0 \cdot 0) \]
\[ = \hat{i}(0 - 24) - \hat{j}(12 - 0) + \hat{k}(24 - 0) \]
\[ = -24\hat{i} - 12\hat{j} + 24\hat{k} \]

### Step 3: Calculate the magnitude of \( \vec{a} \times \vec{b} \)
\[ |\vec{a} \times \vec{b}| = \sqrt{(-24)^2 + (-12)^2 + (24)^2} \]
\[ = \sqrt{576 + 144 + 576} = \sqrt{1296} = 36 \]

### Step 4: Calculate the area of the base of the parallelepiped
The area \( A \) of the base is given by:
\[ A = \frac{1}{2} |\vec{a} \times \vec{b}| = \frac{1}{2} \times 36 = 18 \]

### Step 5: Calculate the height of the parallelepiped
The height \( h \) can be calculated using the projection of \( \vec{c} \) onto \( \vec{a} \times \vec{b} \):
\[ h = \frac{|\vec{c} \cdot (\vec{a} \times \vec{b})|}{|\vec{a} \times \vec{b}|} \]

First, calculate \( \vec{c} \cdot (\vec{a} \times \vec{b}) \):
\[ \vec{c} = 0\hat{i} + 4\hat{j} - 8\hat{k} \]
\[ \vec{c} \cdot (\vec{a} \times \vec{b}) = (0)(-24) + (4)(-12) + (-8)(24) \]
\[ = 0 - 48 - 192 = -240 \]

Taking the absolute value:
\[ |\vec{c} \cdot (\vec{a} \times \vec{b})| = 240 \]

Now, substitute into the height formula:
\[ h = \frac{240}{36} = \frac{20}{3} \]

### Step 6: Calculate the volume of the parallelepiped
The volume \( V \) is given by:
\[ V = A \times h = 18 \times \frac{20}{3} = 120 \]

### Final Answer
The volume of the parallelepiped is \( 120 \).

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