https://file.aiquickdraw.com/mathbot/file/17254544769460uysrcbu.jpg

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?

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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.

Find the constant 'k' so that the vector a = i + 2j - k k is perpendicular to the vector b = 3i - 2j + k.

Learn how to find the constant 'k' such that the vectors a = i + 2j - k k and b = 3i - 2j + k are perpendicular by applying the dot product formula. This problem is ideal for high school students learning vector algebra.

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