The adjacent sides of a parallelogram are given by `vecA=hati+hatj-4hatk` and `vecB=2hati-hatj+4hatk`. Calculate the area of parallelogram.

Answer

To find the area of the parallelogram formed by the vectors \(\vec{A}\) and \(\vec{B}\), we will use the formula for the area of a parallelogram defined by two vectors, which is given by the magnitude of the cross product of the two vectors.

### Step-by-Step Solution:

1. **Identify the Vectors**:
The vectors are given as:
\[ \vec{A} = \hat{i} + \hat{j} - 4\hat{k} \]
\[ \vec{B} = 2\hat{i} - \hat{j} + 4\hat{k} \]

2. **Set Up the Cross Product**:
The area of the parallelogram is given by:
\[ \text{Area} = |\vec{A} \times \vec{B}| \]
To compute \(\vec{A} \times \vec{B}\), we can use the determinant of a matrix formed by the unit vectors and the components of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -4 \\ 2 & -1 & 4 \end{vmatrix} \]

3. **Calculate the Determinant**:
Expanding the determinant:
\[ \vec{A} \times \vec{B} = \hat{i} \begin{vmatrix} 1 & -4 \\ -1 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & -4 \\ 2 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} \]

Now calculating each of the 2x2 determinants:
- For \(\hat{i}\):
\[ = 1 \cdot 4 - (-1) \cdot (-4) = 4 - 4 = 0 \]
- For \(-\hat{j}\):
\[ = 1 \cdot 4 - 2 \cdot (-4) = 4 + 8 = 12 \]
- For \(\hat{k}\):
\[ = 1 \cdot (-1) - 2 \cdot 1 = -1 - 2 = -3 \]

Putting it all together:
\[ \vec{A} \times \vec{B} = 0\hat{i} - 12\hat{j} - 3\hat{k} = -12\hat{j} - 3\hat{k} \]

4. **Find the Magnitude of the Cross Product**:
The magnitude of the vector \(\vec{A} \times \vec{B}\) is calculated as follows:
\[ |\vec{A} \times \vec{B}| = \sqrt{(0)^2 + (-12)^2 + (-3)^2} = \sqrt{0 + 144 + 9} = \sqrt{153} \]

5. **Conclusion**:
Therefore, the area of the parallelogram is:
\[ \text{Area} = \sqrt{153} \text{ square units} \]