The image is slightly blurry, but I'll do my best to extract and interpret the mathematical content from it.

From what I can make out, the problem seems to be asking to "Find the constant 'k' so that the vector $\mathbf{a} = \mathbf{i} + 2\mathbf{j} - k\mathbf{k}$ is perpendicular to the vector $\mathbf{b} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$."

To solve this, we use the concept that two vectors are perpendicular if their dot product is zero.

The dot product of vectors $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$ is given by:

$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$

Here, $\mathbf{a} = \mathbf{i} + 2\mathbf{j} - k\mathbf{k}$ and $\mathbf{b} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}$.

So the dot product is:

$\mathbf{a} \cdot \mathbf{b} = (1)(3) + (2)(-2) + (-k)(1)$

Simplifying:

$\mathbf{a} \cdot \mathbf{b} = 3 - 4 - k = -1 - k$

For the vectors to be perpendicular:

$-1 - k = 0$

Solving for $k$:

$k = -1$

Thus, the constant $k$ is $-1$.

Would you like more details on this problem, or do you have any other questions?

Here are five related questions to deepen your understanding:

1. What is the significance of the dot product in vector analysis?
2. Can two non-zero vectors be perpendicular if one of them is a zero vector?
3. How would you determine if two vectors are parallel instead of perpendicular?
4. What does it mean for two vectors to be orthogonal in 3-dimensional space?
5. How would you find the angle between two vectors using their dot product?

Tip: Always verify your vector components carefully when calculating dot products, as a small mistake can lead to incorrect conclusions.