see attached Certainly! Here is the transcribed text from the image suitable for an educational website:---### Problem StatementWe are given the following vectors:- **Vector \( \mathbf{u} \)**\[ \mathbf{u} = (1, -1, k) \]- **Vector \( \mathbf{v} \)**\[ \mathbf{v} = (k, -1, 2) \]**Question:**What is the value of \( k \) for which the two vectors are orthogonal?---### Explanation:To determine the value of \( k \) for which the two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, we need to recall the definition of orthogonality in terms of the dot product. Two vectors are orthogonal if and only if their dot product is zero.The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is given by:\[ \mathbf{u} \cdot \mathbf{v} = 1 \cdot k + (-1) \cdot (-1) + k \cdot 2 \]Simplifying this,\[ \mathbf{u} \cdot \mathbf{v} = k + 1 + 2k = 3k + 1 \]For the vectors to be orthogonal, the dot product must be zero:\[ 3k + 1 = 0 \]Solving for \( k \):\[ 3k = -1 \]\[ k = -\frac{1}{3} \]Therefore, the value of \( k \) for which the vectors \(\mathbf{u} \) and \(\mathbf{v} \) are orthogonal is \( k = -\frac{1}{3} \).---This solves the problem and illustrates the process of finding the value of \( k \) for orthogonality using the dot product of vectors.

Name: see attached Certainly! Here is the transcribed text from the image su
Uploaded: 2024-03-20T06:46:14.000Z
Channel: Bartleby
Description: see attached Certainly! Here is the transcribed text from the image suitable for an educational website:---### Problem StatementWe are given the following vectors:- **Vector \( \mathbf{u} \)**\[ \mathbf{u} = (1, -1, k) \]- **Vector \( \mathbf{v} \)**