Let x < 0. Find the vector ⃗n =< x, y, z > that is orthogonal to all three vectors (2)
⃗u =< 1, 1, −2 >, ⃗v =< −1, 2, 0 > and w⃗ =< −1, 0, 1 >

Given three vectors \(\mathbf{u} = \langle 1, 1, -2 \rangle\), \(\mathbf{v} = \langle -1, 2, 0 \rangle\), and \(\mathbf{w} = \langle -1, 0, 1 \rangle\), we need to find a vector \(\mathbf{n} = \langle x, y, z \rangle\) that is orthogonal to all three vectors. This means that the dot product of \(\mathbf{n}\) with each of \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) must be zero.

The conditions for orthogonality are:

\[
\mathbf{n} \cdot \mathbf{u} = x(1) + y(1) + z(-2) = 0
\]

\[
\mathbf{n} \cdot \mathbf{v} = x(-1) + y(2) + z(0) = 0
\]

\[
\mathbf{n} \cdot \mathbf{w} = x(-1) + y(0) + z(1) = 0
\]

We can express these equations as:

1. \( x + y - 2z = 0 \)
2. \( -x + 2y = 0 \)
3. \( -x + z = 0 \)

Let's solve this system of linear equations to find \(x\), \(y\), and \(z\).

### Solving the System
From the second equation, we have:
\[
x = 2y
\]

From the third equation, we have:
\[
x = z
\]

Substituting \( x = z \) into the first equation:
\[
z + y - 2z = 0 \implies y = z
\]

Thus, \(x = y = z\).

Given that \(x < 0\), let \(x = -k\), where \(k > 0\). Therefore, the vector \(\mathbf{n}\) is:

\[
\mathbf{n} = \langle -k, -k, -k \rangle
\]

For simplicity, we can choose \(k = 1\), so the vector \(\mathbf{n}\) is:

\[
\mathbf{n} = \langle -1, -1, -1 \rangle
\]

### Final Answer
The vector \(\mathbf{n} = \langle -1, -1, -1 \rangle\) is orthogonal to all three vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\).

Would you like a detailed explanation of the process, or do you have any further questions?

Here are 5 related questions for further exploration:

1. What would happen if \(x > 0\)?
2. How would the solution change if one of the given vectors was different?
3. Can there be other solutions for \(\mathbf{n}\) apart from a scalar multiple?
4. What is the geometric interpretation of this orthogonal vector \(\mathbf{n}\)?
5. How can the cross product be used to find a vector orthogonal to two vectors?

**Tip:** Always check the consistency of the solution with the original conditions, especially when dealing with inequalities like \(x < 0\).

Learn how to find a vector orthogonal to three given vectors (1, 1, -2), (-1, 2, 0), and (-1, 0, 1) in a step-by-step solution. Understand the concept of orthogonality in vector algebra and solve equations to find the orthogonal vector. Suitable for advanced high school students and anyone interested in vector mathematics.

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