To determine the value of $k$ that makes the vectors $\overrightarrow{a} = \left[ 2, k \right]$ and $\overrightarrow{b} = \left[ k, 2 \right]$ parallel, we can use the property that two vectors are parallel if one is a scalar multiple of the other.

We are given the equation:

$\overrightarrow{a} = x \overrightarrow{b}$

where $\overrightarrow{a} = \left[ 2, k \right]$ and $\overrightarrow{b} = \left[ k, 2 \right]$.

This equation translates component-wise into:

[ \left[ 2, k \right] = x \left[ k, 2 \right] ]

Expanding both sides:

1. The first component: $2 = x \cdot k$
2. The second component: $k = x \cdot 2$

Now, solve these two equations for $x$ and $k$.

Step 1: Solve for $x$ in terms of $k$

From the second equation $k = 2x$, we can solve for $x$:

$x = \frac{k}{2}$

Step 2: Substitute into the first equation

Substitute $x = \frac{k}{2}$ into the first equation $2 = x \cdot k$:

$2 = \frac{k}{2} \cdot k$

Simplify:

$2 = \frac{k^2}{2}$

Multiply both sides by 2:

$4 = k^2$

Thus,

$k = \pm 2$

Conclusion

The values of $k$ that make the vectors $\overrightarrow{a} = \left[ 2, k \right]$ and $\overrightarrow{b} = \left[ k, 2 \right]$ parallel are $k = 2$ and $k = -2$.

Would you like further explanation or details on any step? Here are some related questions to explore further:

1. How do we verify if two vectors are parallel using the dot product?
2. What happens if the vectors are not parallel, and how can we check that?
3. How can you find the angle between two vectors geometrically?
4. Can the cross product be used to determine if two vectors are parallel in 3D?
5. What if the vectors were in 3D? How would the conditions for parallelism change?

Tip: To verify if two vectors are parallel, you can check if the determinant of the matrix formed by their components equals zero, or if their dot product is proportional to their magnitudes.