If h is the altitude of a parallelopiped determined by the vectors a,b,c and the base is taken to be the parallelogram determined by a and b where `a= hat(i) +hat(j) + hat(k), b= 2hat(i) + 4hat(j) - hat(k) and c= hat(i) +hat(j) + 3hat(k)`, then the value of `19h^(2)` is

Options

A. 19
B. 16
C. 8
D. None of these

Answer

To solve the problem, we need to find the value of \(19h^2\) where \(h\) is the altitude of the parallelepiped determined by the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\). The base is taken to be the parallelogram determined by \(\mathbf{a}\) and \(\mathbf{b}\).

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

### Step 2: Calculate the cross product \(\mathbf{a} \times \mathbf{b}\)
To find the area of the base (the parallelogram), we need to compute \(\mathbf{a} \times \mathbf{b}\).

Using the determinant form:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 2 & 4 & -1 \end{vmatrix} \]

Calculating the determinant:
\[ = \hat{i} \begin{vmatrix} 1 & 1 \\ 4 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 2 & 4 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
- For \(\hat{i}\):
\[ 1 \cdot (-1) - 1 \cdot 4 = -1 - 4 = -5 \]
- For \(\hat{j}\):
\[ 1 \cdot (-1) - 1 \cdot 2 = -1 - 2 = -3 \quad \text{(remember to change the sign)} \Rightarrow +3 \]
- For \(\hat{k}\):
\[ 1 \cdot 4 - 1 \cdot 2 = 4 - 2 = 2 \]

Thus,
\[ \mathbf{a} \times \mathbf{b} = -5\hat{i} + 3\hat{j} + 2\hat{k} \]

### Step 3: Calculate the magnitude of \(\mathbf{a} \times \mathbf{b}\)
\[ |\mathbf{a} \times \mathbf{b}| = \sqrt{(-5)^2 + 3^2 + 2^2} = \sqrt{25 + 9 + 4} = \sqrt{38} \]

### Step 4: Calculate the volume of the parallelepiped
The volume \(V\) of the parallelepiped is given by:
\[ V = |\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})| \]

Calculating the dot product:
\[ \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) = (\hat{i} + \hat{j} + 3\hat{k}) \cdot (-5\hat{i} + 3\hat{j} + 2\hat{k}) \]
\[ = 1 \cdot (-5) + 1 \cdot 3 + 3 \cdot 2 = -5 + 3 + 6 = 4 \]

Thus, the volume \(V = 4\).

### Step 5: Calculate the height \(h\)
The height \(h\) can be calculated using the formula:
\[ h = \frac{V}{\text{Area of base}} \]
The area of the base is \( |\mathbf{a} \times \mathbf{b}| = \sqrt{38} \).

So,
\[ h = \frac{4}{\sqrt{38}} \]

### Step 6: Calculate \(19h^2\)
Now, we find \(h^2\):
\[ h^2 = \left(\frac{4}{\sqrt{38}}\right)^2 = \frac{16}{38} = \frac{8}{19} \]

Now, calculate \(19h^2\):
\[ 19h^2 = 19 \cdot \frac{8}{19} = 8 \]

### Final Answer
The value of \(19h^2\) is \(8\).