The area of a parallelogram can be calculated using the cross product of two vectors that represent adjacent sides of the parallelogram. If $\mathbf{A} = 2i - j + 3k$ and $\mathbf{B} = i + j$ are the vectors representing the two sides of the parallelogram, the area is given by the magnitude of their cross product:

$\text{Area} = |\mathbf{A} \times \mathbf{B}|$

Step 1: Find the cross product $\mathbf{A} \times \mathbf{B}$

The cross product of two vectors $\mathbf{A} = (2, -1, 3)$ and $\mathbf{B} = (1, 1, 0)$ is calculated as follows:

$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -1 & 3 \\ 1 & 1 & 0 \end{vmatrix}$

Expanding the determinant:

$\mathbf{A} \times \mathbf{B} = \mathbf{i} \begin{vmatrix} -1 & 3 \\ 1 & 0 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & 3 \\ 1 & 0 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix}$

Now, compute each determinant:

$= \mathbf{i} ((-1)(0) - (3)(1)) - \mathbf{j} ((2)(0) - (3)(1)) + \mathbf{k} ((2)(1) - (-1)(1))$ $= \mathbf{i} (0 - 3) - \mathbf{j} (0 - 3) + \mathbf{k} (2 + 1)$ $= -3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}$

Thus, the cross product is:

$\mathbf{A} \times \mathbf{B} = -3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}$

Step 2: Find the magnitude of the cross product

The magnitude of the vector $-3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}$ is:

$|\mathbf{A} \times \mathbf{B}| = \sqrt{(-3)^2 + (3)^2 + (3)^2}$ $= \sqrt{9 + 9 + 9}$ $= \sqrt{27}$ $= 3\sqrt{3}$

Final Answer:

The area of the parallelogram is $3\sqrt{3}$ square units.

Would you like more details or have any questions?

---

Here are 5 related questions to further your understanding:

1. How do you compute the area of a triangle using vector cross products?
2. What is the geometric interpretation of the cross product of two vectors?
3. How do you find the angle between two vectors using the dot product?
4. How can you determine whether two vectors are parallel?
5. What is the formula for the volume of a parallelepiped using vectors?

Tip: The magnitude of the cross product of two vectors gives the area of the parallelogram formed by them, and its direction is perpendicular to the plane containing the two vectors.