If `veca' = hati + hatj, vecb'= hati - hatj + 2hatk` and `vecc' = 2hati - hatj - hatk` then the altitude of the parallelepiped formed by the vectors, `veca, vecb and vecc` having base formed by `vecb and vecc` is ( where `veca'` is recipocal vector `veca) ` `(a)1` `(b)3sqrt2//2` `(c)1//sqrt6` `(d)1//sqrt2`

Options

A. 1
B. `3sqrt2//2`
C. `1//sqrt6`
D. `1//sqrt2`

Explanation

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

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

### Step 2: Calculate the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\)
The volume \(V\) of the parallelepiped can be calculated using the scalar triple product:
\[ V = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

### Step 3: Calculate \(\vec{b} \times \vec{c}\)
To find \(\vec{b} \times \vec{c}\), we can use the determinant of a matrix formed by the unit vectors and the components of \(\vec{b}\) and \(\vec{c}\):
\[ \vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 2 \\ 2 & -1 & -1 \end{vmatrix} \]

Calculating the determinant:
\[ = \hat{i} \begin{vmatrix} -1 & 2 \\ -1 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 2 \\ 2 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -1 \\ 2 & -1 \end{vmatrix} \]

Calculating the minors:
\[ = \hat{i}((-1)(-1) - (2)(-1)) - \hat{j}((1)(-1) - (2)(2)) + \hat{k}((1)(-1) - (-1)(2)) \]
\[ = \hat{i}(1 + 2) - \hat{j}(-1 - 4) + \hat{k}(-1 + 2) \]
\[ = 3\hat{i} + 5\hat{j} + 1\hat{k} \]

So,
\[ \vec{b} \times \vec{c} = 3\hat{i} + 5\hat{j} + 1\hat{k} \]

### Step 4: Calculate the magnitude of \(\vec{b} \times \vec{c}\)
\[ |\vec{b} \times \vec{c}| = \sqrt{3^2 + 5^2 + 1^2} = \sqrt{9 + 25 + 1} = \sqrt{35} \]

### Step 5: Calculate \(\vec{a} \cdot (\vec{b} \times \vec{c})\)
Now we compute:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = (\hat{i} + \hat{j}) \cdot (3\hat{i} + 5\hat{j} + 1\hat{k}) \]
\[ = 1 \cdot 3 + 1 \cdot 5 + 0 \cdot 1 = 3 + 5 = 8 \]

### Step 6: Calculate the altitude
The altitude \(h\) of the parallelepiped is given by:
\[ h = \frac{V}{|\vec{b} \times \vec{c}|} = \frac{8}{\sqrt{35}} \]

### Step 7: Find the correct option
Now we check the options provided:
- (a) 1
- (b) \(\frac{3\sqrt{2}}{2}\)
- (c) \(\frac{1}{\sqrt{6}}\)
- (d) \(\frac{1}{\sqrt{2}}\)

None of the options match our calculated altitude.

### Conclusion
The altitude of the parallelepiped formed by the vectors is \(\frac{8}{\sqrt{35}}\), which does not match any of the provided options.