To solve the problem in the image, we are tasked with finding the area of the parallelogram formed by the vectors $\vec{a} = 2\vec{j} + \vec{k}$ and $\vec{b} = \vec{i} + 2\vec{k}$.

Solution:

The area of a parallelogram formed by two vectors $\vec{a}$ and $\vec{b}$ is given by the magnitude of their cross product:

$\text{Area} = |\vec{a} \times \vec{b}|$

Step 1: Define the vectors

Given the vectors: $\vec{a} = 2\vec{j} + \vec{k}$ $\vec{b} = \vec{i} + 2\vec{k}$

In component form, these vectors can be written as: $\vec{a} = (0, 2, 1)$ $\vec{b} = (1, 0, 2)$

Step 2: Compute the cross product $\vec{a} \times \vec{b}$

The cross product of two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ is given by: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Substitute the components of $\vec{a}$ and $\vec{b}$: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 2 & 1 \\ 1 & 0 & 2 \end{vmatrix}$

Now, expand the determinant: $\vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} 2 & 1 \\ 0 & 2 \end{vmatrix} - \hat{j} \begin{vmatrix} 0 & 1 \\ 1 & 2 \end{vmatrix} + \hat{k} \begin{vmatrix} 0 & 2 \\ 1 & 0 \end{vmatrix}$

$\vec{a} \times \vec{b} = \hat{i} (2 \times 2 - 0 \times 1) - \hat{j} (0 \times 2 - 1 \times 1) + \hat{k} (0 \times 0 - 1 \times 2)$

$\vec{a} \times \vec{b} = \hat{i} (4) - \hat{j} (-1) + \hat{k} (-2)$

Thus, the cross product is: $\vec{a} \times \vec{b} = 4\hat{i} + \hat{j} - 2\hat{k}$ or in component form: $\vec{a} \times \vec{b} = (4, 1, -2)$

Step 3: Compute the magnitude of the cross product

The magnitude of the vector $\vec{a} \times \vec{b} = (4, 1, -2)$ is given by: $|\vec{a} \times \vec{b}| = \sqrt{4^2 + 1^2 + (-2)^2}$

$|\vec{a} \times \vec{b}| = \sqrt{16 + 1 + 4} = \sqrt{21}$

Final Answer:

The area of the parallelogram is $\sqrt{21}$.

Would you like further details or have any questions?

Here are 5 related questions you can explore:

1. How is the cross product of two vectors geometrically interpreted?
2. How do you calculate the volume of a parallelepiped formed by three vectors?
3. What is the difference between the dot product and the cross product of vectors?
4. Can the area of a parallelogram be negative? Why or why not?
5. How does the scalar triple product relate to the cross product?

Tip: Always verify your vector calculations, especially signs, when computing cross products!