https://file.aiquickdraw.com/mathbot/file/1730570000947tvxn8f7s.jpg

To find the area of the parallelogram formed by two vectors \(\vec{u}\) and \(\vec{v}\) as adjacent sides, we need to use the formula:

\[
\text{Area} = \|\vec{u} \times \vec{v}\|
\]

where \(\|\vec{u} \times \vec{v}\|\) is the magnitude of the cross product of \(\vec{u}\) and \(\vec{v}\).

From the image:
- \(\vec{u} = \mathbf{j}\)
- \(\vec{v} = \mathbf{j} + \mathbf{k}\)

### Step 1: Express the vectors in component form
For the basis vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\):
- \(\vec{u} = 0\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} = (0, 1, 0)\)
- \(\vec{v} = 0\mathbf{i} + 1\mathbf{j} + 1\mathbf{k} = (0, 1, 1)\)

### Step 2: Calculate the cross product \(\vec{u} \times \vec{v}\)
The cross product of \(\vec{u} = (0, 1, 0)\) and \(\vec{v} = (0, 1, 1)\) is given by:

\[
\vec{u} \times \vec{v} = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 1 & 0 \\
0 & 1 & 1 \\
\end{vmatrix}
\]

Expanding this determinant:

\[
\vec{u} \times \vec{v} = \mathbf{i}(1 \cdot 1 - 0 \cdot 1) - \mathbf{j}(0 \cdot 1 - 0 \cdot 1) + \mathbf{k}(0 \cdot 1 - 1 \cdot 0)
\]

\[
= \mathbf{i}(1) - \mathbf{j}(0) + \mathbf{k}(0) = \mathbf{i}
\]

So, \(\vec{u} \times \vec{v} = (1, 0, 0)\).

### Step 3: Find the magnitude of the cross product
The magnitude of \((1, 0, 0)\) is:

\[
\|\vec{u} \times \vec{v}\| = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1
\]

### Conclusion
The area of the parallelogram is:

\[
\text{Area} = 1
\]

Would you like further details on any steps or have additional questions?

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### Follow-up Questions:
1. How do we calculate the cross product for vectors in general?
2. Why does the cross product's magnitude represent the area of a parallelogram?
3. What are the conditions under which two vectors are parallel?
4. How does the direction of the cross product relate to the right-hand rule?
5. Can the area ever be negative in this type of calculation?

### Tip:
When working with cross products, remember that only three-dimensional vectors can produce a meaningful cross product.

Find the area of the parallelogram that has the vectors as adjacent sides: u = j, v = j + k.

Learn how to find the area of a parallelogram formed by vectors u = j and v = j + k. This problem covers vector operations, including the cross product formula, and explains how to calculate the area using the magnitude of the cross product.

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